2 dimensional projectile motion (part 1)

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:02.009
Welcome back.

00:00:02.009 --> 00:00:05.580
I will now introduce you to the
concept of-- well, I've

00:00:05.580 --> 00:00:08.130
introduced what vectors are, but
I'll introduce you to how

00:00:08.130 --> 00:00:10.970
they can be even more useful and
we're going to use that to

00:00:10.970 --> 00:00:14.350
actually solve 2-dimensional
projectile motion problems.

00:00:14.350 --> 00:00:16.470
Now we can introduce the game
where you just don't throw the

00:00:16.470 --> 00:00:18.850
ball straight up, you can see
how far you can throw the ball

00:00:18.850 --> 00:00:21.870
or what's the optimal angle
to throw the ball at.

00:00:21.870 --> 00:00:25.220
So hopefully, you'll find this
even more fun, if you can

00:00:25.220 --> 00:00:27.050
imagine it, then the

00:00:27.050 --> 00:00:29.430
1-dimensional projectile motion.

00:00:29.430 --> 00:00:33.450
So everything we've been doing
so far, hopefully, you have an

00:00:33.450 --> 00:00:36.150
intuition that the difference
between a vector and a scalar

00:00:36.150 --> 00:00:38.740
is that a vector
has direction.

00:00:38.740 --> 00:00:47.270
So a vector has-- let me do
colon instead-- has magnitude

00:00:47.270 --> 00:00:48.520
and direction.

00:00:57.100 --> 00:01:02.805
While a scalar has
only magnitude.

00:01:06.420 --> 00:01:10.020
An example of a vector,
velocity is a vector.

00:01:10.020 --> 00:01:11.060
That's easy.

00:01:11.060 --> 00:01:13.820
Maybe we'll start with v, but
velocity is a vector.

00:01:13.820 --> 00:01:17.490
Well in those examples we were
doing before, we said that the

00:01:17.490 --> 00:01:23.640
ball was going-- sometimes we
said that the final velocity

00:01:23.640 --> 00:01:28.360
was minus 30 meters
per second.

00:01:28.360 --> 00:01:30.530
And so where was the magnitude
and what was the direction?

00:01:30.530 --> 00:01:33.590
Well, the magnitude was 30.

00:01:33.590 --> 00:01:36.650
And the direction was-- I knew
what the direction was by the

00:01:36.650 --> 00:01:39.500
sign of that minus sign.

00:01:39.500 --> 00:01:42.960
In that same problem, if someone
asked me, what is the

00:01:42.960 --> 00:01:47.610
speed of the ball right before
it hits the ground?

00:01:47.610 --> 00:01:50.240
I would've said oh, the speed is
just 30 meters per second.

00:01:50.240 --> 00:01:52.660
And I wouldn't have to say 30
meters per second down or

00:01:52.660 --> 00:01:53.920
minus 30 per second.

00:01:53.920 --> 00:01:55.230
I'd just say 30 meters
per second

00:01:55.230 --> 00:01:56.310
because that's its speed.

00:01:56.310 --> 00:01:58.250
I don't know what direction
it's going in that speed.

00:01:58.250 --> 00:01:59.370
So that's its speed.

00:01:59.370 --> 00:02:04.990
And in a general way, in one
dimension, you could say that

00:02:04.990 --> 00:02:10.350
the scalar of the same quantity
tends to be the

00:02:10.350 --> 00:02:16.260
absolute value of the
vector quantity.

00:02:16.260 --> 00:02:18.420
Because in this example the
speed is just the absolute

00:02:18.420 --> 00:02:21.570
value of this.

00:02:21.570 --> 00:02:26.640
If my initial velocity was
plus-- that's not equal,

00:02:26.640 --> 00:02:28.070
that's actually just equal.

00:02:28.070 --> 00:02:30.810
If the initial velocity was
plus 30 meters per second,

00:02:30.810 --> 00:02:33.610
then the speed would still
be 30 meters per second.

00:02:33.610 --> 00:02:35.350
So I just wanted to introduce
that notion.

00:02:35.350 --> 00:02:40.470
And in one dimension, velocity
can kind of be-- direction can

00:02:40.470 --> 00:02:43.380
be implied by positive
or negative numbers.

00:02:43.380 --> 00:02:46.720
But now, when we go into two
dimensions, we're going to

00:02:46.720 --> 00:02:49.050
have to get a little bit more
sophisticated in terms of how

00:02:49.050 --> 00:02:52.650
we give direction.

00:02:52.650 --> 00:02:54.110
So let me clear this.

00:02:58.410 --> 00:03:03.510
Let me give an example
where I take a ball.

00:03:03.510 --> 00:03:10.630
I take a green ball, clearly a
green ball, and I throw it--

00:03:10.630 --> 00:03:12.670
this is a straight line.

00:03:12.670 --> 00:03:19.280
I throw it at 10 meters per
second, but I don't throw it

00:03:19.280 --> 00:03:19.965
straight up.

00:03:19.965 --> 00:03:25.600
I actually throw it at a-- let
me think of a good angle.

00:03:25.600 --> 00:03:27.960
I throw it at a--
I don't know.

00:03:27.960 --> 00:03:31.110
I throw it at a 45
degree angle.

00:03:31.110 --> 00:03:32.590
Or if you like, radians,
that's what?

00:03:32.590 --> 00:03:36.220
So that's pi, pi over 4.

00:03:36.220 --> 00:03:38.770
Let's say I throw it at
a 45 degree angle.

00:03:38.770 --> 00:03:41.300
So now my problem becomes
a lot more complicated.

00:03:41.300 --> 00:03:45.400
One, how do I specify the
magnitude and direction?

00:03:45.400 --> 00:03:47.840
And then, how do I solve
this problem?

00:03:47.840 --> 00:03:50.520
How do I figure out how high
it goes, how far it goes.

00:03:50.520 --> 00:03:51.480
et cetera.

00:03:51.480 --> 00:03:56.400
And hopefully, that's what we
will be able to teach you over

00:03:56.400 --> 00:03:58.730
the course of the next
couple of videos.

00:03:58.730 --> 00:04:01.310
This is the ground, right?

00:04:01.310 --> 00:04:03.730
So one, if I just want to give
the speed and direction, or

00:04:03.730 --> 00:04:06.030
the magnitude and the direction
for this vector

00:04:06.030 --> 00:04:08.210
quantity, I have just
specified it.

00:04:08.210 --> 00:04:13.820
You know, I could to say my
velocity or my initial

00:04:13.820 --> 00:04:22.610
velocity is 10 meters per second
at 45 degrees to the

00:04:22.610 --> 00:04:24.000
horizon or something
like that.

00:04:24.000 --> 00:04:28.560
So that would be a legitimate
velocity specification.

00:04:28.560 --> 00:04:30.810
But it's kind of hard to deal
with, so what I'm going to

00:04:30.810 --> 00:04:32.740
teach you is something very
neat about vectors.

00:04:32.740 --> 00:04:38.410
Is that you can actually break
up a vector into-- I don't

00:04:38.410 --> 00:04:40.670
want to introduce two
complicated terms, but into

00:04:40.670 --> 00:04:42.550
its orthogonal components.

00:04:42.550 --> 00:04:46.890
So you can say well, something
moving at 10 meters per second

00:04:46.890 --> 00:04:55.300
at this angle, is actually
moving at some speed upwards

00:04:55.300 --> 00:04:59.950
and it's moving at some speed,
in this case, to the right.

00:04:59.950 --> 00:05:02.100
Let's say in the positive
direction.

00:05:02.100 --> 00:05:05.400
So if you imagine this
coordinate axis, it's moving

00:05:05.400 --> 00:05:10.780
in the y direction at some speed
and it's moving in the x

00:05:10.780 --> 00:05:12.430
direction at some speed.

00:05:12.430 --> 00:05:17.110
And when they're combined at
some way, you get this angle

00:05:17.110 --> 00:05:19.590
or you're moving at kind
of an incline.

00:05:19.590 --> 00:05:23.290
So how do I figure out-- if I
have 10 meters per second and

00:05:23.290 --> 00:05:27.340
I'm moving-- how do I break
that down into my vertical

00:05:27.340 --> 00:05:30.720
speed and my horizontal speed?

00:05:30.720 --> 00:05:33.750
And I'll show you that it's
actually pretty cool.

00:05:33.750 --> 00:05:35.325
It's pretty straightforward.

00:05:37.850 --> 00:05:41.410
What you can do is you can
actually draw a vector where

00:05:41.410 --> 00:05:46.700
the length of the vector
is its magnitude.

00:05:46.700 --> 00:05:48.510
Let's draw it again.

00:05:48.510 --> 00:05:50.680
So let's say this is a
coordinate axis and that the

00:05:50.680 --> 00:05:54.700
length of this line
right here is 10.

00:05:54.700 --> 00:05:58.200
The length of the this line is
10 and it represents speed or

00:05:58.200 --> 00:06:00.280
velocity at this point.

00:06:00.280 --> 00:06:04.640
And we know this angle
is 45 degrees.

00:06:04.640 --> 00:06:11.110
So it turns out that the
vertical component of the

00:06:11.110 --> 00:06:15.800
ball's velocity would be
equivalent to the length of

00:06:15.800 --> 00:06:19.870
this side, and the horizontal
component would be equivalent

00:06:19.870 --> 00:06:21.120
to the length of this side.

00:06:25.860 --> 00:06:28.010
This arrow right here is the
same thing as this arrow, I

00:06:28.010 --> 00:06:30.130
just translated it
to the right.

00:06:30.130 --> 00:06:33.440
Just so that you see that
it's just a triangle.

00:06:33.440 --> 00:06:35.030
And it's just not any triangle,

00:06:35.030 --> 00:06:38.060
it's a right triangle.

00:06:38.060 --> 00:06:41.680
And what's useful-- I don't want
to get too complicated

00:06:41.680 --> 00:06:42.530
with linear algebra.

00:06:42.530 --> 00:06:44.790
But it's important that the
two directions that you're

00:06:44.790 --> 00:06:48.380
breaking it up in are-- well,
they don't have to be, but it

00:06:48.380 --> 00:06:50.720
makes it useful that they are at
right angles to each other

00:06:50.720 --> 00:06:53.410
because then we can start to use
a lot of our trig ratios.

00:06:53.410 --> 00:06:55.360
So let's just figure
out this problem.

00:06:55.360 --> 00:06:56.840
I think you just need to
see a lot of example.

00:06:56.840 --> 00:07:00.840
So we're going 10 meters per
second at 45 degrees.

00:07:00.840 --> 00:07:05.150
Well, what speed am I going
straight up then?

00:07:05.150 --> 00:07:08.060
What is the straight up
component of this 10 meters

00:07:08.060 --> 00:07:09.390
per second?

00:07:09.390 --> 00:07:11.570
Well, how do we figure out this
side if this was just a

00:07:11.570 --> 00:07:13.720
regular triangle?

00:07:13.720 --> 00:07:15.270
If this is the hypotenuse,
this is

00:07:15.270 --> 00:07:17.610
opposite and this is adjacent.

00:07:17.610 --> 00:07:21.366
So let me break out
some trigonometry.

00:07:21.366 --> 00:07:23.430
soh cah toa.

00:07:23.430 --> 00:07:25.430
And if you don't remember your
trig or you never learned

00:07:25.430 --> 00:07:29.330
trig, I encourage you to go
watch the trig videos that we

00:07:29.330 --> 00:07:32.320
have. soh cah toa.

00:07:32.320 --> 00:07:36.900
Which of the trig ratios involve
the opposite and the

00:07:36.900 --> 00:07:38.150
hypotenuse?

00:07:40.490 --> 00:07:42.780
So our sine.

00:07:42.780 --> 00:07:48.630
And we know that sine of theta
is equal to opposite over

00:07:48.630 --> 00:07:49.820
hypotenuse.

00:07:49.820 --> 00:07:57.720
So we know that the sine of 45
degrees is equal to-- let's

00:07:57.720 --> 00:08:04.410
call this my velocity
of vertical.

00:08:04.410 --> 00:08:06.370
I've never seen that variable
used in actual physics

00:08:06.370 --> 00:08:08.030
classes, but I think
it makes sense.

00:08:08.030 --> 00:08:11.890
This is the velocity vertical
and this is the velocity

00:08:11.890 --> 00:08:12.920
horizontal.

00:08:12.920 --> 00:08:16.830
And later I'll introduce more
conventional conventions, but

00:08:16.830 --> 00:08:18.010
that'll do us for now.

00:08:18.010 --> 00:08:21.810
So my vertical velocity, the
velocity that I'm going

00:08:21.810 --> 00:08:27.380
straight up is-- well, sine of
45 degrees is my vertical

00:08:27.380 --> 00:08:31.250
velocity over my kind of
angled velocity, this

00:08:31.250 --> 00:08:34.799
hypotenuse over 10.

00:08:34.799 --> 00:08:36.710
And what's my horizontal
velocity?

00:08:36.710 --> 00:08:39.260
And so if I were to solve for
vertical velocity, I'd get

00:08:39.260 --> 00:08:43.760
vertical velocity is
equal to 10 times

00:08:43.760 --> 00:08:48.660
the sine of 45 degrees.

00:08:48.660 --> 00:08:51.760
And then similarly, my
horizontal velocity, this is

00:08:51.760 --> 00:08:55.340
the adjacent over hypotenuse--
cah.

00:08:55.340 --> 00:08:57.250
So it's cosine.

00:08:57.250 --> 00:09:02.510
So I could say cosine of 45
degrees is equal to my

00:09:02.510 --> 00:09:05.350
horizontal velocity over 10.

00:09:05.350 --> 00:09:12.220
Or my horizontal velocity is
equal to 10 times the cosine

00:09:12.220 --> 00:09:15.030
of 45 degrees.

00:09:15.030 --> 00:09:16.970
I'm about to run out of time, so
I'm going to continue this

00:09:16.970 --> 00:09:18.600
in the next video.


Projectile motion (part 10)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.130
I'm now just going to review
a couple of more projectile

00:00:03.130 --> 00:00:05.230
motion problems, because I
really want you to understand

00:00:05.230 --> 00:00:06.380
this as deeply as possible.

00:00:06.380 --> 00:00:08.630
Because the rest of physics
that we do and the rest of

00:00:08.630 --> 00:00:12.120
mechanics, a lot of it really
just depends on your

00:00:12.120 --> 00:00:14.280
understanding of projectile
motion.

00:00:14.280 --> 00:00:15.810
And hopefully this will
be intuitive for

00:00:15.810 --> 00:00:16.620
you after some point.

00:00:16.620 --> 00:00:18.760
Let's just go back to a
problem we did before.

00:00:18.760 --> 00:00:21.820
Let's say that I'm at the top
of a building and I take a

00:00:21.820 --> 00:00:24.570
rock or a ball-- and I don't
recommend that you do this--

00:00:24.570 --> 00:00:27.000
and I drop it.

00:00:27.000 --> 00:00:30.010
I dropped the ball.

00:00:30.010 --> 00:00:39.890
And let's say that I know that
the building is, well I don't

00:00:39.890 --> 00:00:43.530
know, let's say it's a
very tall building.

00:00:43.530 --> 00:00:45.250
Well let's say it's
not too tall.

00:00:45.250 --> 00:00:47.600
Let's say we're throwing
it from the 10th story.

00:00:47.600 --> 00:00:53.280
So it's roughly 100 feet, or
that's about 30 meters, right?

00:00:53.280 --> 00:00:55.950
So the ball is going to go 30
meters down, so the change in

00:00:55.950 --> 00:01:02.520
distance is going to be minus
30 meters, right?

00:01:02.520 --> 00:01:04.230
The ball is going to
fall 30 meters.

00:01:04.230 --> 00:01:09.740
Now if I ask you, hey you, how
fast is that ball going to be

00:01:09.740 --> 00:01:12.050
right before it hits
the ground?

00:01:12.050 --> 00:01:16.180
Well I could rewrite all the
formulas, but I think

00:01:16.180 --> 00:01:19.230
hopefully you know them by now,
but there's two kind of

00:01:19.230 --> 00:01:21.080
interesting formulas
we might use.

00:01:21.080 --> 00:01:23.230
I'll also show you maybe how to
do this more intuitively.

00:01:23.230 --> 00:01:29.320
But there's vf squared is equal
to vi squared plus 2ad.

00:01:29.320 --> 00:01:34.680
And then there is and this is
2 times delta d, right?

00:01:34.680 --> 00:01:38.690
Whoops, plus 2a times
change in distance.

00:01:38.690 --> 00:01:40.870
Sometimes I'll just write
that as distance.

00:01:40.870 --> 00:01:44.260
And then there was change in
distance is equal to the

00:01:44.260 --> 00:01:48.980
initial velocity times time
plus acceleration time

00:01:48.980 --> 00:01:51.920
squared over 2.

00:01:51.920 --> 00:01:53.310
And the there were
the other ones.

00:01:53.310 --> 00:01:57.300
Change in distance is rate times
time, and velocity is

00:01:57.300 --> 00:01:59.620
acceleration times time, and
you know about all average

00:01:59.620 --> 00:02:01.700
velocity, et cetera,
et cetera.

00:02:01.700 --> 00:02:04.570
So one of these formulas I
think might strike you as

00:02:04.570 --> 00:02:06.700
immediately useful to what we're
trying to figure out,

00:02:06.700 --> 00:02:09.389
because we're trying to figure
out what the velocity is when

00:02:09.389 --> 00:02:10.449
you hit the ground, right?

00:02:10.449 --> 00:02:13.460
We want to know what the
final velocity is.

00:02:13.460 --> 00:02:17.220
We know the initial velocity
is 0, right?

00:02:17.220 --> 00:02:19.180
We're just letting it go.

00:02:19.180 --> 00:02:20.870
And then what's the
acceleration?

00:02:20.870 --> 00:02:27.100
Well the acceleration is
gravity, which is equal to

00:02:27.100 --> 00:02:30.110
minus 10 meters per
second squared.

00:02:30.110 --> 00:02:33.620
So we could use this formula,
cause we know vi, we know

00:02:33.620 --> 00:02:36.050
acceleration, we know change in
distance, so we could solve

00:02:36.050 --> 00:02:38.740
for velocity.

00:02:38.740 --> 00:02:42.860
So we get vf squared is equal
to vi squared-- well that's

00:02:42.860 --> 00:02:49.360
just 0, right-- times 2
acceleration, and so

00:02:49.360 --> 00:02:51.560
that's minus 10.

00:02:51.560 --> 00:02:54.110
Then what's the distance?

00:02:54.110 --> 00:02:56.700
Well the distance is
minus 30, right?

00:02:56.700 --> 00:03:00.000
Change in distance
is minus 30.

00:03:00.000 --> 00:03:06.130
So you get vf squared is equal
to-- 10 times 30 is 300 is

00:03:06.130 --> 00:03:17.940
equal to 600, so if I took 600
square root-- roughly 24.5

00:03:17.940 --> 00:03:20.520
meters per second.

00:03:20.520 --> 00:03:30.060
So vf is equal to approximately

00:03:30.060 --> 00:03:34.360
24.5 meters per second.

00:03:34.360 --> 00:03:37.310
Now if we wanted to know well
how long did it did take the

00:03:37.310 --> 00:03:40.920
thing to hit the ground, or how
long should it take the

00:03:40.920 --> 00:03:41.990
thing to hit the ground.

00:03:41.990 --> 00:03:45.180
Well then we could use
this formula, right?

00:03:45.180 --> 00:03:47.460
Cause we know the change in
distance, we know the initial

00:03:47.460 --> 00:03:50.410
velocity is 0, we know the
acceleration, and so we could

00:03:50.410 --> 00:03:52.300
just solve for t.

00:03:52.300 --> 00:03:58.100
So change in distance I said
was minus 30, right?

00:03:58.100 --> 00:04:00.190
The ball went down 30 meters.

00:04:00.190 --> 00:04:03.320
We could say minus 30.

00:04:03.320 --> 00:04:05.210
Let me change the colors, cause
I want to make sure you

00:04:05.210 --> 00:04:06.460
know which formula I'm using.

00:04:12.120 --> 00:04:13.700
I'll do it in that color.

00:04:13.700 --> 00:04:17.209
Minus 30 is equal to the initial
velocity of 0-- so the

00:04:17.209 --> 00:04:19.630
initial velocity times anything
is 0, so this term

00:04:19.630 --> 00:04:23.680
cancels out-- plus at
squared over 2.

00:04:23.680 --> 00:04:30.740
So what's a? a is minus 10
divided by 2 is minus 5.

00:04:30.740 --> 00:04:34.430
So it's minus 5 t squared.

00:04:34.430 --> 00:04:38.260
And so you divide both sides by
5, and you get t squared is

00:04:38.260 --> 00:04:40.230
equal to 6.

00:04:40.230 --> 00:04:41.920
Once again use the trusty
calculator.

00:04:41.920 --> 00:04:46.780
The square root of 6 is
roughly 2.45 seconds.

00:04:46.780 --> 00:04:52.100
t is equal to 2.45 seconds.

00:04:52.100 --> 00:04:52.790
It's pretty neat.

00:04:52.790 --> 00:04:56.190
Using these two formulas
in about 3 minutes.

00:04:56.190 --> 00:04:58.390
We probably could do it faster
if we weren't explaining it,

00:04:58.390 --> 00:05:01.490
so you could do it really
fast on an exam.

00:05:01.490 --> 00:05:06.760
I just said I drop a ball from
a building at 30 meters high.

00:05:06.760 --> 00:05:09.530
Very quickly, we were able to
figure out how fast it was

00:05:09.530 --> 00:05:12.920
falling when it hit the ground
and how long should it take

00:05:12.920 --> 00:05:14.570
for it to fall.

00:05:14.570 --> 00:05:16.300
And if I were to give you one
of the other pieces of

00:05:16.300 --> 00:05:17.090
information.

00:05:17.090 --> 00:05:23.320
If I told you that the ball took
2.45 seconds to fall, and

00:05:23.320 --> 00:05:24.830
I didn't tell you the distance,
then you could have

00:05:24.830 --> 00:05:27.080
figured out the distance based
on that information using the

00:05:27.080 --> 00:05:28.290
same formula.

00:05:28.290 --> 00:05:33.390
Hopefully, that makes
some sense.

00:05:38.310 --> 00:05:41.270
Actually, I'll leave
it there right now.

00:05:41.270 --> 00:05:43.770
I just wanted to show you that
when you actually just go

00:05:43.770 --> 00:05:46.080
straight to the formulas, and
you don't do all the kind of

00:05:46.080 --> 00:05:48.030
meandering explanations
that I tend to do,

00:05:48.030 --> 00:05:49.010
which I do for a reason.

00:05:49.010 --> 00:05:50.295
I do them to give you
an intuition.

00:05:50.295 --> 00:05:54.570
But when you just go straight
to the formulas, these

00:05:54.570 --> 00:05:58.250
problems don't really
take much time.

00:05:58.250 --> 00:05:59.490
But with that said, let me
show you how to do it

00:05:59.490 --> 00:06:02.810
intuitively if you forget
the formulas.

00:06:02.810 --> 00:06:07.330
So once again, I'm
dropping a ball.

00:06:07.330 --> 00:06:10.840
It's a 30 meter building,
right?

00:06:10.840 --> 00:06:15.530
And I want to figure out what
the speed is when it hits and

00:06:15.530 --> 00:06:16.730
how long did it take to fall?

00:06:16.730 --> 00:06:18.050
Well let's just start
with our basics.

00:06:18.050 --> 00:06:20.620
Let's say I just had amnesia,
and the only thing I remember

00:06:20.620 --> 00:06:26.910
is change in distance is equal
to rate times time, or average

00:06:26.910 --> 00:06:28.930
rate or average velocity
times time.

00:06:28.930 --> 00:06:34.920
And I know that change in
velocity is equal to

00:06:34.920 --> 00:06:39.010
acceleration times time,
if I know that my

00:06:39.010 --> 00:06:40.490
acceleration is constant.

00:06:40.490 --> 00:06:44.100
Otherwise, it would be the
average acceleration.

00:06:44.100 --> 00:06:45.600
But this is all you
have to know.

00:06:45.600 --> 00:06:47.040
And I'm going to show
you right now.

00:06:47.040 --> 00:06:50.655
So we know what the change
in distance is, right?

00:06:50.655 --> 00:06:52.750
And just so not to confuse you,
we're doing the exact

00:06:52.750 --> 00:06:53.300
same problem.

00:06:53.300 --> 00:06:54.555
But I'm just showing you that
you didn't have to memorize

00:06:54.555 --> 00:06:57.340
the formulas, although it might
be faster if you do.

00:06:57.340 --> 00:07:00.790
So the change in distance is
minus 30 meters, right?

00:07:00.790 --> 00:07:01.900
So let's plug that
in straight.

00:07:01.900 --> 00:07:06.375
So we know that minus 30
meters-- this is what I'm

00:07:06.375 --> 00:07:11.220
going to use-- is equal
to the average

00:07:11.220 --> 00:07:17.240
velocity times time, right?

00:07:17.240 --> 00:07:20.620
Well what's the average
velocity?

00:07:20.620 --> 00:07:22.210
Let's see if we can
figure that out.

00:07:22.210 --> 00:07:24.220
Well we know the average
velocity is just the average

00:07:24.220 --> 00:07:26.900
of the initial and the
final velocity.

00:07:26.900 --> 00:07:30.330
Well what's the initial
velocity?

00:07:30.330 --> 00:07:34.320
The initial velocity is 0.

00:07:34.320 --> 00:07:36.070
vi is 0.

00:07:36.070 --> 00:07:39.210
The final velocity is
the final velocity.

00:07:39.210 --> 00:07:40.770
We don't know that yet.

00:07:40.770 --> 00:07:42.550
So what's the average velocity
going to be?

00:07:42.550 --> 00:07:45.140
It's going to be the average
of these 2 numbers.

00:07:45.140 --> 00:07:47.880
So the average velocity is going
to be-- this a little

00:07:47.880 --> 00:07:53.060
too bloody, I don't like this
color-- the final velocity

00:07:53.060 --> 00:07:56.630
plus the initial velocity
over 2 which

00:07:56.630 --> 00:07:59.700
equals vf over 2, right?

00:07:59.700 --> 00:08:03.120
So let's put that into
our formula.

00:08:03.120 --> 00:08:08.020
So minus 30 is equal to
the average velocity.

00:08:08.020 --> 00:08:12.870
That's just the final velocity
divided by 2 times time.

00:08:12.870 --> 00:08:14.260
And now you're saying,
but Sal, we

00:08:14.260 --> 00:08:15.170
still have 2 variables.

00:08:15.170 --> 00:08:18.135
We can't solve for either
one of them, right?

00:08:18.135 --> 00:08:25.400
Well if we could write time in
terms of final velocity, then

00:08:25.400 --> 00:08:26.550
we could solve for
final velocity.

00:08:26.550 --> 00:08:28.540
So let's try doing that.

00:08:28.540 --> 00:08:32.880
So let's use this equation
right there.

00:08:32.880 --> 00:08:35.760
Change in velocity is equal to
acceleration times time.

00:08:35.760 --> 00:08:37.080
What's the change in velocity?

00:08:37.080 --> 00:08:43.100
It's the final velocity minus
the initial velocity, and that

00:08:43.100 --> 00:08:44.540
equals acceleration.

00:08:44.540 --> 00:08:47.360
Acceleration is minus 10 meters
per second squared

00:08:47.360 --> 00:08:48.500
times time.

00:08:48.500 --> 00:08:49.880
And what's the initial
velocity?

00:08:49.880 --> 00:08:52.020
That's 0, right?

00:08:52.020 --> 00:08:57.020
So we know that vf-- this is
0-- is equal to minus 10t.

00:08:57.020 --> 00:08:59.640
And if we wanted to write the
other way, we could say t is

00:08:59.640 --> 00:09:05.860
equal to minus vf over 10.

00:09:05.860 --> 00:09:08.540
I just rearranged
this equation.

00:09:08.540 --> 00:09:10.930
Now I could take this
and substitute it

00:09:10.930 --> 00:09:13.520
back into this equation.

00:09:13.520 --> 00:09:24.600
And I get minus 30 is equal to
vf over 2 times t, which is

00:09:24.600 --> 00:09:26.320
this right here.

00:09:26.320 --> 00:09:31.460
So times minus vf over 10.

00:09:31.460 --> 00:09:37.790
And if you solve this, you get,
well you essentially the

00:09:37.790 --> 00:09:40.050
exact thing same thing we got
in the previous problem.

00:09:40.050 --> 00:09:42.210
You get vf squared
is equal to 600.

00:09:42.210 --> 00:09:44.610
I'm skipping some steps just
cause I'm running out of time.

00:09:44.610 --> 00:09:47.560
And you get the same answer,
whatever, 24.5 seconds.

00:09:47.560 --> 00:09:49.745
I just wanted to do that,
because I wanted to show that

00:09:49.745 --> 00:09:53.310
you really just have to memorize
these two things.

00:09:53.310 --> 00:09:55.670
And then you can derive all
those equations on the fly,

00:09:55.670 --> 00:09:58.210
although it can be faster to
know them ahead of time.

00:09:58.210 --> 00:10:00.790
But these are the only
two things that

00:10:00.790 --> 00:10:02.150
we're ever really using.

00:10:02.150 --> 00:10:04.870
Those other equations are just
derivations of them.

00:10:04.870 --> 00:10:06.120
I'll see you in the
next presentation.


Projectile motion (part 9)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.450
Hello.

00:00:01.450 --> 00:00:05.380
So the last couple of videos,
well we've been doing a lot of

00:00:05.380 --> 00:00:06.760
projectile motion, and
we'll keep doing

00:00:06.760 --> 00:00:08.029
some projectile motion.

00:00:08.029 --> 00:00:12.150
But I think it may have seemed
more confusing than it really

00:00:12.150 --> 00:00:14.010
is, because I kept everything
in terms of variables.

00:00:14.010 --> 00:00:16.810
And we worked with the formulas
and the variables.

00:00:16.810 --> 00:00:20.910
What I want to do is take all
those things that hopefully

00:00:20.910 --> 00:00:23.580
I've taught you or you've had
exposure to over the last

00:00:23.580 --> 00:00:25.850
couple of videos, and just do a
bunch of problems. Because I

00:00:25.850 --> 00:00:28.150
think this is all about
experience and getting the

00:00:28.150 --> 00:00:30.750
intuition and just checking
through a lot of problems.

00:00:30.750 --> 00:00:34.110
So let's just do a
straightforward one, kind of

00:00:34.110 --> 00:00:36.070
going back to the ball
throwing game.

00:00:36.070 --> 00:00:43.366
So let's say I throw a ball
straight up, throw a ball up,

00:00:43.366 --> 00:00:49.200
and that the total time the ball
is in the air is, I don't

00:00:49.200 --> 00:00:51.070
know, let's say it's
7 seconds.

00:00:54.762 --> 00:01:04.120
So how fast did I throw
it straight up?

00:01:04.120 --> 00:01:06.130
Not throw up.

00:01:06.130 --> 00:01:08.200
How fast did I throw
the ball up?

00:01:08.200 --> 00:01:10.380
And then how high did it go?

00:01:18.250 --> 00:01:20.400
We've figured out a bunch of
formulas, and we could plug it

00:01:20.400 --> 00:01:22.200
straight into those formula and
solve this problem in 10

00:01:22.200 --> 00:01:24.800
seconds, or probably more
like 2 seconds.

00:01:24.800 --> 00:01:27.400
But what I want to do is just
actually go through the same

00:01:27.400 --> 00:01:29.270
logic we went through to get
the formulas, but I'll work

00:01:29.270 --> 00:01:29.840
with the numbers.

00:01:29.840 --> 00:01:31.230
And I'll show you this isn't
something where you even have

00:01:31.230 --> 00:01:32.530
to memorize formulas.

00:01:32.530 --> 00:01:34.990
So it took 7 seconds, right?

00:01:34.990 --> 00:01:40.160
The ball took 7 seconds to go
all the way up and then to go

00:01:40.160 --> 00:01:41.780
down, right?

00:01:41.780 --> 00:01:44.220
And assuming no air resistance,
it took the same

00:01:44.220 --> 00:01:46.260
amount of time to go
to its peak as it

00:01:46.260 --> 00:01:48.570
took to go down, right?

00:01:48.570 --> 00:01:56.760
So it took 3.5 seconds
to go up.

00:02:01.170 --> 00:02:04.700
Time is 3.5 seconds.

00:02:04.700 --> 00:02:07.330
And then it took another
3.5 seconds to go

00:02:07.330 --> 00:02:11.720
down, right, 3.5.

00:02:11.720 --> 00:02:13.640
So it took 3.5 seconds to
get to its peak, so what

00:02:13.640 --> 00:02:14.685
does that tell me?

00:02:14.685 --> 00:02:19.240
Well it tells me that in 3.5
seconds, the ball went from

00:02:19.240 --> 00:02:21.905
whatever the initial velocity
it was when it left my hand.

00:02:21.905 --> 00:02:30.350
The ball went from-- v sub i is
equal to whatever-- it went

00:02:30.350 --> 00:02:33.080
from that initial velocity.

00:02:33.080 --> 00:02:33.670
This is the ball.

00:02:33.670 --> 00:02:35.190
This isn't a number.

00:02:35.190 --> 00:02:38.290
And then by the time it got to
the top, to kind of the peak

00:02:38.290 --> 00:02:42.050
of its travel, it got
to 0 velocity.

00:02:42.050 --> 00:02:45.660
So I'll call that v sub f
or the final velocity.

00:02:45.660 --> 00:02:48.060
It went from whatever
speed I threw it

00:02:48.060 --> 00:02:50.755
at to 0 in 3.5 seconds.

00:02:53.860 --> 00:02:56.030
So now we could just use,
I mean we don't even

00:02:56.030 --> 00:02:56.670
have to use a formula.

00:02:56.670 --> 00:02:58.400
We could think of
it intuitively.

00:02:58.400 --> 00:03:02.010
How fast am I decelerating?

00:03:02.010 --> 00:03:07.470
So I'm decelerating at the
rate of gravity, right?

00:03:07.470 --> 00:03:09.510
And we know that cause that's
what's slowing down the ball.

00:03:09.510 --> 00:03:13.450
So we know that acceleration
is equal to minus 10 meters

00:03:13.450 --> 00:03:15.700
per second squared.

00:03:15.700 --> 00:03:16.900
And so what do we do?

00:03:16.900 --> 00:03:19.735
Well acceleration is the same
thing as a change in velocity.

00:03:23.460 --> 00:03:25.385
Acceleration times time
is the same thing as

00:03:25.385 --> 00:03:26.590
the change in velocity.

00:03:26.590 --> 00:03:29.010
Or the change in velocity
divided by time is

00:03:29.010 --> 00:03:30.610
acceleration.

00:03:30.610 --> 00:03:31.150
So let's say that.

00:03:31.150 --> 00:03:31.875
It's a change in velocity.

00:03:31.875 --> 00:03:38.510
Well the change in velocity is
equal to the final velocity

00:03:38.510 --> 00:03:40.580
minus the initial velocity.

00:03:40.580 --> 00:03:42.710
And that equals acceleration
times time.

00:03:42.710 --> 00:03:50.040
So minus 10 meters per second
squared-- I'll stick with the

00:03:50.040 --> 00:03:51.480
units this time-- times time.

00:03:51.480 --> 00:03:56.710
So it did that in 3-1/2 seconds,
times 3.5 seconds.

00:03:56.710 --> 00:03:59.600
So this seconds cancels out with
one of the seconds in the

00:03:59.600 --> 00:04:01.970
denominator here.

00:04:01.970 --> 00:04:13.020
So I get minus vi is equal to
minus 350 meters per second

00:04:13.020 --> 00:04:16.540
minus 10 times, no, that's
3.5, not 35.

00:04:16.540 --> 00:04:19.130
So it's minus 35 meters
per second.

00:04:19.130 --> 00:04:22.410
Something was looking
suspicious.

00:04:22.410 --> 00:04:25.240
Minus 35 meters per second, and
then of course, multiply

00:04:25.240 --> 00:04:26.600
both sides by negative 1.

00:04:26.600 --> 00:04:31.490
And the initial velocity is
equal to 35 meters per second.

00:04:31.490 --> 00:04:32.520
That wasn't too bad.

00:04:32.520 --> 00:04:34.810
You don't even have to really
do it formally like this.

00:04:34.810 --> 00:04:38.720
You could just say well I'm
throwing it at some speed.

00:04:38.720 --> 00:04:40.820
Of the total time, it takes
half of that time

00:04:40.820 --> 00:04:42.290
for it go to 0, right?

00:04:42.290 --> 00:04:45.060
So I threw it at some speed and
it took 3-1/2 seconds to

00:04:45.060 --> 00:04:50.200
go to 0, and it's decelerating
at this speed, right?

00:04:50.200 --> 00:04:54.400
So every second it's getting 10
meters per second slower.

00:04:54.400 --> 00:04:56.520
Every second I go,
it's getting 10

00:04:56.520 --> 00:04:58.070
meters per second slower.

00:04:58.070 --> 00:05:04.040
So if it took 3-1/2 seconds to
go to 0, I must have started

00:05:04.040 --> 00:05:06.480
off at 35 meters per second.

00:05:06.480 --> 00:05:08.880
If that confuses you, use the
formula, but I want to give

00:05:08.880 --> 00:05:10.270
you a sense that you can
kind of just use

00:05:10.270 --> 00:05:11.440
your intuition, right?

00:05:11.440 --> 00:05:16.710
Every second it travels in the
air, it's getting 10 meters

00:05:16.710 --> 00:05:18.160
per second slower.

00:05:18.160 --> 00:05:22.610
It took 3-1/2 seconds to go from
whatever speed you threw

00:05:22.610 --> 00:05:28.780
it at to 0, so it must have
started off at 35 meters per

00:05:28.780 --> 00:05:30.200
second, right?

00:05:30.200 --> 00:05:34.600
At t equals 1, it was going
25 meters per second.

00:05:34.600 --> 00:05:38.990
At t equals 2, it was going
15 meters per second.

00:05:38.990 --> 00:05:44.570
At t equals 3, the ball is going
straight up at 5 meters

00:05:44.570 --> 00:05:45.590
per second.

00:05:45.590 --> 00:05:48.400
And at t equals 3.5, the
ball is stationary.

00:05:48.400 --> 00:05:49.910
It's right at the top
and then it starts

00:05:49.910 --> 00:05:51.740
decelerating downwards.

00:05:51.740 --> 00:05:56.372
Let's do some more problems.
I don't like these colors.

00:05:56.372 --> 00:05:59.623
Let me switch to something
nicer.

00:06:04.220 --> 00:06:06.980
Also in that problem, I didn't
figure out how high the ball

00:06:06.980 --> 00:06:13.920
went, so once again the total
travel time is 7 seconds.

00:06:13.920 --> 00:06:16.040
We know the acceleration
is minus 10

00:06:16.040 --> 00:06:18.480
meters per second squared.

00:06:18.480 --> 00:06:20.890
And we know that the initial
velocity now-- we figured it

00:06:20.890 --> 00:06:24.540
out-- is 35 meters per second.

00:06:24.540 --> 00:06:26.380
So to figure out how high
it went, so let's

00:06:26.380 --> 00:06:27.750
just draw that part.

00:06:27.750 --> 00:06:30.830
So the ball is going
up, right?

00:06:30.830 --> 00:06:36.110
And we figured out just in its
upward portion to go from vi

00:06:36.110 --> 00:06:41.900
is equal to 35 meters per second
to, let's call this vf,

00:06:41.900 --> 00:06:44.510
is equal to 0 meters
per second.

00:06:44.510 --> 00:06:51.980
It took the ball 3.5 seconds,
right, because this is half of

00:06:51.980 --> 00:06:52.670
its travel time.

00:06:52.670 --> 00:06:55.920
And then it took another 3.5
seconds to come straight down.

00:06:55.920 --> 00:06:58.930
Well we could just use our
fairly straightforward formula

00:06:58.930 --> 00:07:04.290
that change in distance is equal
to the average velocity

00:07:04.290 --> 00:07:06.460
times time, right?

00:07:06.460 --> 00:07:08.430
And this should be pretty
intuitive to you.

00:07:08.430 --> 00:07:09.840
This shouldn't even be
viewed as a formula.

00:07:09.840 --> 00:07:12.370
Hopefully, it's starting
to be viewed as

00:07:12.370 --> 00:07:14.760
maybe even common sense.

00:07:14.760 --> 00:07:16.030
So what was the average
velocity?

00:07:16.030 --> 00:07:18.240
The average velocity is just
the average of your initial

00:07:18.240 --> 00:07:21.710
velocity and your velocity
at the top of the curve.

00:07:21.710 --> 00:07:23.320
So that's 0.

00:07:23.320 --> 00:07:26.917
So the average of 35 and 0 is
just 35 divided by 2, so it

00:07:26.917 --> 00:07:30.960
equals 35 over 2.

00:07:30.960 --> 00:07:32.850
And then what's the time?

00:07:32.850 --> 00:07:34.100
3.5 seconds.

00:07:36.690 --> 00:07:38.490
And I could use my trusty

00:07:38.490 --> 00:07:41.290
calculator, provided by Windows.

00:07:41.290 --> 00:07:58.418
35 divided by 2 times 3.5 is
equal to 61.25 meters.

00:07:58.418 --> 00:08:01.440
It's pretty easy.

00:08:01.440 --> 00:08:03.710
So hopefully you're realizing
that we used a lot of formulas

00:08:03.710 --> 00:08:04.180
and things.

00:08:04.180 --> 00:08:07.290
And the formulas were getting
all over the page last time.

00:08:07.290 --> 00:08:11.930
But when you actually do it with
numbers, it's not so bad.

00:08:11.930 --> 00:08:14.570
In the next video now, I'm going
to just keep doing a

00:08:14.570 --> 00:08:16.640
bunch more problems, because I
think you're probably tired of

00:08:16.640 --> 00:08:19.625
this ball throwing business.

00:08:19.625 --> 00:08:20.875
I'll see you soon.


Projectile motion (part 6)

WEBVTT
Kind: captions
Language: en

00:00:01.000 --> 00:00:01.910
Welcome back.

00:00:01.910 --> 00:00:04.330
Actually, before I teach you how
to figure out how high the

00:00:04.330 --> 00:00:06.300
ball went-- and you might
already be able to figure it

00:00:06.300 --> 00:00:08.450
out-- I want to show you
really kind of a more

00:00:08.450 --> 00:00:11.080
intuitive way for figuring out
how fast the ball went.

00:00:11.080 --> 00:00:13.470
I used the equation this first
time, just to really show you

00:00:13.470 --> 00:00:17.810
that this equation can be
useful, but I personally

00:00:17.810 --> 00:00:21.430
always forget equations, so I
find it very useful to have

00:00:21.430 --> 00:00:23.750
kind of a common sense way of
figuring it out, as well.

00:00:23.750 --> 00:00:26.650
And really, this equation was
derived by coming up with a

00:00:26.650 --> 00:00:29.440
common sense way-- I don't know
if that last statement

00:00:29.440 --> 00:00:31.440
made sense, but let's move on.

00:00:31.440 --> 00:00:34.200
Let's say that same problem, but
let's just think about it

00:00:34.200 --> 00:00:37.170
without our equations, because
that's always a good fall back

00:00:37.170 --> 00:00:39.520
when you're panicking in the
middle of the exam, and you

00:00:39.520 --> 00:00:43.190
can't remember if an equation
had a 1/2, or a 2, or a minus,

00:00:43.190 --> 00:00:46.820
or a plus, or t, or t squared.

00:00:46.820 --> 00:00:49.470
It's good just to think about
what's happening.

00:00:49.470 --> 00:00:53.770
So when I throw a ball straight
up-- I have a, let's

00:00:53.770 --> 00:00:56.770
say it's a baseball, it looks
like a baseball-- and I throw

00:00:56.770 --> 00:01:04.050
it straight up where my velocity
initial is equal to--

00:01:04.050 --> 00:01:06.790
well, let's say that
this the variable.

00:01:06.790 --> 00:01:09.300
This is my initial velocity,
v sub i.

00:01:09.300 --> 00:01:10.380
What's going to happen?

00:01:10.380 --> 00:01:12.250
As soon as I throw it up,
it's going to start

00:01:12.250 --> 00:01:13.760
decellerating, right?

00:01:13.760 --> 00:01:19.110
Because I have the force of
gravity decelerating it

00:01:19.110 --> 00:01:22.660
immediately-- so, with gravity,
we're saying minus 10

00:01:22.660 --> 00:01:26.640
meters per second
squared, right?

00:01:26.640 --> 00:01:30.360
This ball is going to keep
decelerating until its

00:01:30.360 --> 00:01:32.560
velocity goes to 0, right?

00:01:32.560 --> 00:01:40.460
The ball, if we were to graph
time and distance, where this

00:01:40.460 --> 00:01:44.182
is time, and then this
is distance.

00:01:44.182 --> 00:01:44.600
Right?

00:01:44.600 --> 00:01:47.620
At time zero, we're
on the ground.

00:01:47.620 --> 00:01:52.510
The ball starts off going
really fast, then starts

00:01:52.510 --> 00:01:57.170
slowing down, then its velocity
goes to 0, and then

00:01:57.170 --> 00:01:59.856
it starts accelerating in the
negative direction, and it

00:01:59.856 --> 00:02:02.890
starts going fast, and bam--
it hits the ground again.

00:02:02.890 --> 00:02:05.340
What happens is that the ball
starts fast, and starts going

00:02:05.340 --> 00:02:08.699
slower, slower, slower, slower,
and slower, until its

00:02:08.699 --> 00:02:15.180
velocity is 0, and then it
starts-- you could say

00:02:15.180 --> 00:02:17.750
reaccelerating in the opposite
direction-- or decellerating,

00:02:17.750 --> 00:02:20.185
really-- but reaccelerating in
the opposite direction, and

00:02:20.185 --> 00:02:21.520
then it hits the ground.

00:02:21.520 --> 00:02:26.350
Actually, we know-- assuming
nothing about air resistance,

00:02:26.350 --> 00:02:30.170
et cetera et cetera-- that the
velocity that it hits the

00:02:30.170 --> 00:02:32.390
ground with is the same velocity
that it left your

00:02:32.390 --> 00:02:35.220
hand with, just in the
opposite direction.

00:02:35.220 --> 00:02:37.410
So there's a couple of
interesting things here.

00:02:37.410 --> 00:02:43.300
The time at which its velocity
is at 0-- so that point right

00:02:43.300 --> 00:02:49.080
there-- that's going to
be at t equals 2.

00:02:49.080 --> 00:02:52.410
We know that this shape is
actually a parabola, if you

00:02:52.410 --> 00:02:53.430
remember that from
algebra two.

00:02:53.430 --> 00:02:54.350
Why is that a parabola?

00:02:54.350 --> 00:02:55.460
What was the equation for it?

00:02:55.460 --> 00:02:57.300
We figured out the equation
using that previous formula.

00:02:57.300 --> 00:02:59.510
I don't want to use it this
time, but what was that

00:02:59.510 --> 00:03:00.490
previous formula?

00:03:00.490 --> 00:03:06.120
It was change in distance
is equal to vit plus at

00:03:06.120 --> 00:03:07.230
squared over 2.

00:03:07.230 --> 00:03:09.570
It's a parabola, but I think if
you had thought about it,

00:03:09.570 --> 00:03:11.850
you would have realized
also it's a parabola.

00:03:11.850 --> 00:03:15.050
And it points downward because
a is negative, so the t

00:03:15.050 --> 00:03:18.710
squared term is negative--
that's why it opens to the

00:03:18.710 --> 00:03:20.166
downward side.

00:03:20.166 --> 00:03:21.850
I think that might make
a little sense to you.

00:03:21.850 --> 00:03:25.600
So what we could figure out
is, if we're given a t, we

00:03:25.600 --> 00:03:30.200
could say well, half of that
number-- let's say that t

00:03:30.200 --> 00:03:34.230
equals 10 seconds.

00:03:34.230 --> 00:03:37.380
So we know that in 10 seconds,
the ball left my hand, went up

00:03:37.380 --> 00:03:40.650
some distance, and then came
back down and hit the ground.

00:03:40.650 --> 00:03:46.490
What we also know then, though,
is that t over 2 at 5

00:03:46.490 --> 00:03:49.860
seconds, the ball was
essentially stationary for

00:03:49.860 --> 00:03:52.890
just a moment-- its velocity had
decelerated, decelerated,

00:03:52.890 --> 00:03:57.560
decelerated, and hit 0, and then
right before it started

00:03:57.560 --> 00:04:00.670
reaccelerating again, or
reaccelerating downwards, its

00:04:00.670 --> 00:04:03.835
velocity was 0 at the
time t equals 0.

00:04:07.820 --> 00:04:10.340
The fact that the ball
decelerated from my initial

00:04:10.340 --> 00:04:15.510
velocity to 0 in 5 seconds--
what does that tell us?

00:04:15.510 --> 00:04:20.360
Well, we have the very simple
equation, you know, change in

00:04:20.360 --> 00:04:22.960
velocity is equal to

00:04:22.960 --> 00:04:26.380
acceleration times time, right?

00:04:26.380 --> 00:04:29.010
You probably knew that before
watching any of these videos.

00:04:29.010 --> 00:04:30.615
And the change in the
acceleration, well, that's

00:04:30.615 --> 00:04:35.690
just the final velocity minus
the initial velocity is equal

00:04:35.690 --> 00:04:38.460
to the acceleration
times time.

00:04:38.460 --> 00:04:40.610
In this situation, what's
the final velocity?

00:04:40.610 --> 00:04:43.120
Remember, we're not going to
go all the way here-- we're

00:04:43.120 --> 00:04:46.900
just figuring out from here
to time equals 2, right?

00:04:46.900 --> 00:04:48.900
So what's the final velocity?

00:04:48.900 --> 00:04:51.870
We're saying that point is where
the ball is not going

00:04:51.870 --> 00:04:54.865
up, and it's not going down,
so its final velocity is 0.

00:04:54.865 --> 00:04:59.160
So, 0 minus initial velocity
is equal to acceleration--

00:04:59.160 --> 00:05:01.700
acceleration is the acceleration
of gravity--

00:05:01.700 --> 00:05:05.805
minus 10 meters per
second squared.

00:05:13.090 --> 00:05:14.100
I know it's a little confusing,
because I'm using

00:05:14.100 --> 00:05:20.610
the same t, but let's say that
this time is t sub 0, just to

00:05:20.610 --> 00:05:23.490
kind of make sure it's not a
variable, it's actual time.

00:05:23.490 --> 00:05:26.820
So this is t sub naught
over 2, right?

00:05:26.820 --> 00:05:31.480
Because the ball is motionless
right at the peak of its--

00:05:31.480 --> 00:05:35.700
we're not an arc, but right
at the peak of its travel.

00:05:35.700 --> 00:05:39.370
So, it's acceleration times
time, but at this time, the

00:05:39.370 --> 00:05:41.365
time is going to be t
sub naught over 2.

00:05:47.120 --> 00:05:51.980
Once again, the 0 doesn't
matter, and we can multiply

00:05:51.980 --> 00:05:53.920
both sides times negative 1.

00:05:53.920 --> 00:06:00.290
We get plus vi, and we get vi is
equal to 10 divided by 2--

00:06:00.290 --> 00:06:05.165
5 meters per second squared
t sub naught.

00:06:05.165 --> 00:06:09.730
Which was exactly what we got in
the previous video when we

00:06:09.730 --> 00:06:10.800
used this formula.

00:06:10.800 --> 00:06:14.000
And I think it makes sense to
you -- hopefully, this was

00:06:14.000 --> 00:06:17.620
kind of an intuitive way of
thinking about what happened.

00:06:17.620 --> 00:06:19.850
Before actually I do the
distance, I actually want to

00:06:19.850 --> 00:06:22.080
graph what's happening, because
I think it just dawned

00:06:22.080 --> 00:06:24.620
on me that that might be
something that will give you

00:06:24.620 --> 00:06:25.770
more intuition.

00:06:25.770 --> 00:06:27.785
I'm all about giving you
intuition so you never forget

00:06:27.785 --> 00:06:30.030
this stuff.

00:06:30.030 --> 00:06:34.140
So this is, if we were to
graph-- that's an ugly looking

00:06:34.140 --> 00:06:37.800
axes, but I think you'll get the
point-- this is distance,

00:06:37.800 --> 00:06:40.000
this is time.

00:06:40.000 --> 00:06:43.486
We already said it's going to be
like a parabola, like that.

00:06:43.486 --> 00:06:44.890
Right?

00:06:44.890 --> 00:06:50.610
Where this is t sub naught over
2, this t sub naught.

00:06:50.610 --> 00:06:54.330
It launches really fast, then
it slows down, and then it's

00:06:54.330 --> 00:06:56.360
motionless right here,
and then it starts

00:06:56.360 --> 00:06:57.580
reaccelerating downwards.

00:06:57.580 --> 00:06:59.470
If that's the distance,
what does the

00:06:59.470 --> 00:07:02.790
velocity graph look like?

00:07:02.790 --> 00:07:05.530
The velocity graph, I'll draw
right below-- I'll draw it in

00:07:05.530 --> 00:07:09.144
another color, just
for variety.

00:07:09.144 --> 00:07:10.394
That's a bold.

00:07:13.840 --> 00:07:17.440
So over-- actually, that's not
how I want to draw it.

00:07:17.440 --> 00:07:20.400
I have to draw the negative
side, too.

00:07:25.060 --> 00:07:33.950
So this is time, and then this
axis is velocity, so we start

00:07:33.950 --> 00:07:35.480
off at a positive
velocity, right?

00:07:35.480 --> 00:07:43.450
We start off at v sub i, and
what's going to happen here is

00:07:43.450 --> 00:07:46.180
the velocity decreases at
a constant rate, right?

00:07:46.180 --> 00:07:48.797
And that rate is just the
rate of acceleration.

00:07:48.797 --> 00:07:57.440
The velocity decreases until
at t sub naught-- let me

00:07:57.440 --> 00:08:03.190
switch back to yellow-- at t sub
naught-- woops, I'm using

00:08:03.190 --> 00:08:04.850
the wrong tool, it actually
looked like I was drawing

00:08:04.850 --> 00:08:10.740
something-- at t sub naught,
the velocity now is

00:08:10.740 --> 00:08:12.130
negative vi, right?

00:08:12.130 --> 00:08:14.930
Remember we said, when the ball
comes back down, it's

00:08:14.930 --> 00:08:16.500
going at the same velocity,
just at

00:08:16.500 --> 00:08:18.150
the opposite direction.

00:08:18.150 --> 00:08:23.040
This point right here, which is
t sub naught over 2, that

00:08:23.040 --> 00:08:25.560
corresponds to this
point, right?

00:08:25.560 --> 00:08:28.550
Which makes sense, because
that's the point at which the

00:08:28.550 --> 00:08:29.915
ball has no velocity.

00:08:29.915 --> 00:08:32.200
And look, the velocity is 0.

00:08:32.200 --> 00:08:36.100
So the ball starts going up
really fast, slows down at a

00:08:36.100 --> 00:08:38.960
constant rate, and what is
the slope of this line?

00:08:38.960 --> 00:08:41.130
Well, the slope is just the
acceleration, right?

00:08:41.130 --> 00:08:43.419
Because velocity is the
acceleration times time.

00:08:43.419 --> 00:08:46.390
And then it's stationary for
just a moment, because its

00:08:46.390 --> 00:08:51.060
velocity is 0, and then it
starts accelerating-- or you

00:08:51.060 --> 00:08:53.150
could say decelerating, or
accelerating in the negative

00:08:53.150 --> 00:08:54.780
direction-- until the
point that it's

00:08:54.780 --> 00:08:58.240
going at v sub i down.

00:08:58.240 --> 00:09:02.634
And of course, if you were to
graph acceleration-- if I were

00:09:02.634 --> 00:09:06.540
to graph acceleration
over time--

00:09:06.540 --> 00:09:10.930
acceleration is constant.

00:09:10.930 --> 00:09:11.705
It's right here.

00:09:11.705 --> 00:09:14.890
Let me just get a line tool.

00:09:14.890 --> 00:09:17.850
Acceleration is just a constant
minus 10 meters per

00:09:17.850 --> 00:09:19.420
second, so it's going
to look like that.

00:09:19.420 --> 00:09:21.420
And it's just the slope
of this line.

00:09:21.420 --> 00:09:25.370
If you know calculus, it'll make
sense to you that this

00:09:25.370 --> 00:09:28.475
line is the derivative of this
line, or this curve.

00:09:28.475 --> 00:09:30.570
This line is the derivative
of this curve.

00:09:30.570 --> 00:09:32.990
And even if you don't know
calculus, I think it makes

00:09:32.990 --> 00:09:38.010
sense to you that this is
the slope of this line.

00:09:38.010 --> 00:09:39.920
And just so, if you haven't
learned calculus, a derivative

00:09:39.920 --> 00:09:43.460
is just to a way of figuring out
a slope at any point along

00:09:43.460 --> 00:09:45.770
the curve, so it's nothing
too fancy.

00:09:45.770 --> 00:09:47.770
I'll see you in the
next presentation.


Projectile motion (part 8)

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.330
Welcome back.

00:00:01.330 --> 00:00:04.280
I got too excited, and I went 20
seconds over time-- I hope

00:00:04.280 --> 00:00:06.960
YouTube still lets me publish
this video, so I'll try my

00:00:06.960 --> 00:00:07.790
best.

00:00:07.790 --> 00:00:12.710
What I was saying is how related
these two formulas

00:00:12.710 --> 00:00:15.885
are, and that makes sense
because at this point the ball

00:00:15.885 --> 00:00:19.060
is stationary-- it's completely
the same as a ball

00:00:19.060 --> 00:00:22.090
just being dropped, a stationary
ball being dropped

00:00:22.090 --> 00:00:23.710
from height h.

00:00:23.710 --> 00:00:26.100
We can use this formula to do
the same thing that we did

00:00:26.100 --> 00:00:29.010
intuitively in the last video.

00:00:29.010 --> 00:00:31.875
If we're just dropping a ball,
the initial velocity is zero,

00:00:31.875 --> 00:00:34.110
so this term goes to zero,
and this term is the

00:00:34.110 --> 00:00:36.480
only one that matters.

00:00:36.480 --> 00:00:39.150
Once again, if we're dropping
it, we started at height h and

00:00:39.150 --> 00:00:42.900
we go to zero, so the change
in distance is now minus h.

00:00:42.900 --> 00:00:46.780
We have at squared over 2, and
that equals gravity times--

00:00:46.780 --> 00:00:48.000
what's the time?

00:00:48.000 --> 00:00:56.960
The time is not this full t0,
it's t0 over 2, so g t0 over 2

00:00:56.960 --> 00:00:59.170
squared over 2.

00:00:59.170 --> 00:01:01.780
You multiply both sides times
negative 1, and you get this

00:01:01.780 --> 00:01:03.610
exact same thing.

00:01:03.610 --> 00:01:04.530
That's pretty cool.

00:01:04.530 --> 00:01:07.720
What I love about physics it
that there's so many ways to

00:01:07.720 --> 00:01:10.280
approach the problems. You can
actually start off with any

00:01:10.280 --> 00:01:13.720
one of the formulas that we've
been looking at, and as long

00:01:13.720 --> 00:01:15.990
as you kind of keep playing
with it and keep thinking

00:01:15.990 --> 00:01:18.680
about it, you might not find
the fastest route to the

00:01:18.680 --> 00:01:21.390
solution, but as long as you
don't do anything incorrect,

00:01:21.390 --> 00:01:23.330
you will get the solution.

00:01:23.330 --> 00:01:25.950
Hopefully, this gives you a
little bit of an intuition of

00:01:25.950 --> 00:01:27.230
projectile motion.

00:01:27.230 --> 00:01:30.400
In the next set of videos,
we're going to make it

00:01:30.400 --> 00:01:33.970
slightly more complicated, but
also slightly more realistic.

00:01:33.970 --> 00:01:35.700
I'm going to introduce you
to what happens in

00:01:35.700 --> 00:01:37.850
two-dimensional motion-- it's
when you're not throwing

00:01:37.850 --> 00:01:40.360
something straight up, but you
actually throw something at an

00:01:40.360 --> 00:01:43.080
angle like you normally do.

00:01:43.080 --> 00:01:46.400
I'll see you in the
next video.


Projectile motion (part 7)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.680
Welcome back.

00:00:01.680 --> 00:00:05.470
Now I think that hopefully you
have some intuition, and maybe

00:00:05.470 --> 00:00:07.090
you can figure this out
on your own, but I'll

00:00:07.090 --> 00:00:07.760
show how to do it.

00:00:07.760 --> 00:00:10.920
We spent a lot of time on how
fast did I throw the ball--

00:00:10.920 --> 00:00:13.050
how fast was it when it left my
hand-- so let's figure out

00:00:13.050 --> 00:00:14.970
how high did it go, because
that's kind of a cool thing to

00:00:14.970 --> 00:00:17.110
know, as well.

00:00:17.110 --> 00:00:23.070
We figured out that the final
velocity-- actually, the

00:00:23.070 --> 00:00:27.840
initial velocity-- is just
equal to minus the

00:00:27.840 --> 00:00:33.530
acceleration of gravity
times time over 2.

00:00:33.530 --> 00:00:35.650
If we say that the acceleration
of gravity is

00:00:35.650 --> 00:00:38.530
minus 10 meters per second
squared, then this becomes

00:00:38.530 --> 00:00:43.970
5t-- so that's initial velocity,
and the final

00:00:43.970 --> 00:00:46.695
velocity is just the negative
of the initial velocity.

00:00:46.695 --> 00:00:50.140
It's going back to what
we were doing before.

00:00:50.140 --> 00:00:51.260
The initial velocity--
we want to figure out

00:00:51.260 --> 00:00:53.140
how high did we go?

00:00:53.140 --> 00:00:55.970
Let's just go back to our
basic-- we want to figure out

00:00:55.970 --> 00:00:57.700
this point, and we want
to figure out how high

00:00:57.700 --> 00:00:59.820
did this thing go.

00:00:59.820 --> 00:01:01.650
That's actually the for vertex
of this parabola, and if you

00:01:01.650 --> 00:01:04.739
know the formula for a vertex of
parabola, you could do it--

00:01:04.739 --> 00:01:07.010
once again, I like to do it
intuitively, as opposed to

00:01:07.010 --> 00:01:08.440
memorizing formulas.

00:01:08.440 --> 00:01:10.200
We want to figure out how
high did this go--

00:01:10.200 --> 00:01:12.440
let's call that h.

00:01:12.440 --> 00:01:14.030
So what happens?

00:01:14.030 --> 00:01:20.880
I'm going from an initial
velocity of vi, and I'm going

00:01:20.880 --> 00:01:24.620
to a final velocity-- well, not
a final velocity, you can

00:01:24.620 --> 00:01:28.940
call this an intermediate
velocity-- a velocity of 0.

00:01:28.940 --> 00:01:33.020
It takes a time of t sub naught
over 2-- that doesn't

00:01:33.020 --> 00:01:44.660
look too good-- it takes
t over 2, or t

00:01:44.660 --> 00:01:46.150
sub naught over 2.

00:01:46.150 --> 00:01:48.520
I know it gets confusing when
it's t sub naught-- t sub

00:01:48.520 --> 00:01:51.260
naught just says this is an
actual constant t as opposed

00:01:51.260 --> 00:01:53.220
to just the variable t,
and that's all it is.

00:01:53.220 --> 00:01:55.810
I know it can get confusing
sometimes.

00:01:55.810 --> 00:01:59.710
So, the initial velocity is
equal to minus g times

00:01:59.710 --> 00:02:00.990
whatever we time, or
whatever time your

00:02:00.990 --> 00:02:02.790
friend gets on the timer.

00:02:02.790 --> 00:02:08.060
But we also know that between
time 0 and t sub naught over

00:02:08.060 --> 00:02:11.350
2-- so half the time that your
friend has on the timer-- the

00:02:11.350 --> 00:02:16.100
velocity went from this
velocity to 0.

00:02:16.100 --> 00:02:20.070
The average velocity over that
time-- not the average

00:02:20.070 --> 00:02:22.780
velocity over the entire time,
the average velocity over just

00:02:22.780 --> 00:02:30.650
this first interval-- the
average velocity here would

00:02:30.650 --> 00:02:39.770
just be v sub i plus 0, because
we're kind of this

00:02:39.770 --> 00:02:45.070
intermediate velocity, because
we're stationary up here, so

00:02:45.070 --> 00:02:48.440
we divide that by 2.

00:02:48.440 --> 00:02:51.760
v sub i is just this, so if we
take this, and substitute

00:02:51.760 --> 00:02:59.270
here, we get their average
velocity over this time

00:02:59.270 --> 00:03:04.810
interval is equal to
minus g times t sub

00:03:04.810 --> 00:03:08.440
naught over 4, now.

00:03:08.440 --> 00:03:11.830
Remember, this is only the
average velocity over this

00:03:11.830 --> 00:03:14.760
time period-- the average
velocity over the entire time

00:03:14.760 --> 00:03:18.310
period over t sub naught is
actually 0, because we start

00:03:18.310 --> 00:03:20.200
and end at the same place.

00:03:20.200 --> 00:03:22.630
This is kind of an interesting
point as to how speed and

00:03:22.630 --> 00:03:24.250
velocity is different.

00:03:24.250 --> 00:03:25.560
You're probably thinking--
hey, Sal, how could the

00:03:25.560 --> 00:03:28.530
average velocity over the entire
time be 0, when the

00:03:28.530 --> 00:03:32.450
ball was clearly going quite
fast most of the time?

00:03:32.450 --> 00:03:33.820
That's the difference between
speed and velocity.

00:03:33.820 --> 00:03:36.960
It turns out that the speed was,
I agree with you, quite

00:03:36.960 --> 00:03:41.080
fast the entire time, but when
you take the average velocity,

00:03:41.080 --> 00:03:44.490
all the times that the velocity
was positive, is

00:03:44.490 --> 00:03:47.760
balanced off by the times that
the velocity is negative.

00:03:47.760 --> 00:03:52.580
Over the entire period, the
average of the velocity is 0,

00:03:52.580 --> 00:03:56.960
and that makes sense because,
the ball didn't go anywhere--

00:03:56.960 --> 00:03:59.500
it just came back to where it
started, and so that's an

00:03:59.500 --> 00:04:03.030
interesting distinction between
speed and velocity.

00:04:03.030 --> 00:04:04.840
Anyway-- going back, and I hope
I'm not confusing you--

00:04:04.840 --> 00:04:11.440
the average velocity over this
time period is minus g times t

00:04:11.440 --> 00:04:12.540
sub naught over 4.

00:04:12.540 --> 00:04:15.250
If we were to graph it here,
that would be right here-- it

00:04:15.250 --> 00:04:21.574
would be essentially this,
at this point.

00:04:24.930 --> 00:04:29.140
This is our average velocity
over half the time period, and

00:04:29.140 --> 00:04:31.270
it would be equal to
v sub i over 2,

00:04:31.270 --> 00:04:34.050
which is equal to this.

00:04:34.050 --> 00:04:37.360
We know our average velocity
over this time period, and we

00:04:37.360 --> 00:04:40.500
also know the time-- the time is
t sub naught over 2, so we

00:04:40.500 --> 00:04:44.710
can just say change in distance
is equal to average

00:04:44.710 --> 00:04:49.030
velocity times time.

00:04:49.030 --> 00:04:51.610
What's the average velocity?

00:04:51.610 --> 00:05:05.660
It's all this stuff minus
gt sub naught over 4.

00:05:09.970 --> 00:05:15.090
The average velocity is this,
and what's the time?

00:05:15.090 --> 00:05:17.250
We're saying the average
velocity over t sub naught

00:05:17.250 --> 00:05:21.960
over 2, not the whole t sub
naught-- maybe if I do some

00:05:21.960 --> 00:05:24.625
examples with numbers, this
won't be as confusing.

00:05:24.625 --> 00:05:28.150
So, the time is t sub
naught over 2.

00:05:28.150 --> 00:05:35.350
So t sub naught over 2.

00:05:35.350 --> 00:05:37.470
That's the time.

00:05:37.470 --> 00:05:43.235
And so the distance we traveled,
h, is equal to-- if

00:05:43.235 --> 00:05:54.840
we multiply it out-- minus gt
sub naught squared over 8.

00:05:54.840 --> 00:05:56.170
That's interesting.

00:05:56.170 --> 00:06:00.950
If I throw a ball, and so this
is h-- I know I'm making this

00:06:00.950 --> 00:06:09.360
really crowded-- if I throw a
ball, and it takes 5 seconds

00:06:09.360 --> 00:06:14.560
to go all the way up, and
go all the way down.

00:06:14.560 --> 00:06:17.330
If I just substituted in this
equation so that it takes 5

00:06:17.330 --> 00:06:22.420
seconds to go all the way up,
and all the way down-- once

00:06:22.420 --> 00:06:25.950
again, this is minus g, so if
gravity is actually a minus 10

00:06:25.950 --> 00:06:29.190
meters per second, so this
becomes a positive.

00:06:29.190 --> 00:06:34.840
Going back, if I say h is
equal to-- so minus g is

00:06:34.840 --> 00:06:38.850
positive 10 meters per second
times 5 seconds.

00:06:41.725 --> 00:06:52.630
It's 10 times 5 over 8, so
40 over 8 equals 50--

00:06:52.630 --> 00:06:53.755
oh, sorry, t squared.

00:06:53.755 --> 00:06:55.180
I was about to say something,
because that's

00:06:55.180 --> 00:06:58.300
not right-- t squared.

00:06:58.300 --> 00:07:02.230
It's minus g t naught
squared over 8.

00:07:02.230 --> 00:07:04.050
So what's 10 times 25?

00:07:04.050 --> 00:07:13.880
It's 250 over 8, so that's
30 something meters--

00:07:13.880 --> 00:07:15.430
it's like 31 meters.

00:07:15.430 --> 00:07:18.420
So if I can throw a ball, and
it stays in the air for a

00:07:18.420 --> 00:07:22.150
total of 5 seconds, I threw it
31 meters into the air-- that

00:07:22.150 --> 00:07:28.280
might not sound high, but
31 meters is 93 feet.

00:07:28.280 --> 00:07:30.490
That's equivalent to a nine
story building, so that

00:07:30.490 --> 00:07:32.120
actually is pretty high.

00:07:32.120 --> 00:07:36.260
If you can if you can keep a
ball in the air for 5 seconds,

00:07:36.260 --> 00:07:38.240
you're pretty strong.

00:07:38.240 --> 00:07:39.800
I want to show you something
else that's kind of

00:07:39.800 --> 00:07:42.020
interesting, or at least
interesting to me.

00:07:42.020 --> 00:07:43.880
Let me erase some of this,
just because I think it's

00:07:43.880 --> 00:07:46.315
getting really crazy now.

00:07:46.315 --> 00:07:48.200
Let me erase this stuff.

00:07:54.164 --> 00:07:58.030
See, I'll even erase
this stuff.

00:07:58.030 --> 00:08:00.340
I think you get it now, and
you can just back up and

00:08:00.340 --> 00:08:02.990
rewatch it-- that's the beauty
of these videos.

00:08:02.990 --> 00:08:06.710
You don't really have to take
notes, but just back it up and

00:08:06.710 --> 00:08:07.960
rewatch the video.

00:08:11.650 --> 00:08:12.230
OK.

00:08:12.230 --> 00:08:15.350
I think I have a clean
space now.

00:08:15.350 --> 00:08:18.080
So let me just do a
nice color, and

00:08:18.080 --> 00:08:20.260
back to the pen tool.

00:08:20.260 --> 00:08:28.250
We got that the height is
equal to minus gt naught

00:08:28.250 --> 00:08:35.059
squared over 8, but we could
have also written this as--

00:08:35.059 --> 00:08:37.710
I'm just playing around with
this algebraically-- this is

00:08:37.710 --> 00:08:49.870
the same thing as minus g over
2 times t naught squared-- t

00:08:49.870 --> 00:08:51.370
naught over 2 squared.

00:08:51.370 --> 00:08:54.220
This is just the exact
same thing.

00:08:54.220 --> 00:09:00.030
I was wondering, does this
concept look familiar?

00:09:00.030 --> 00:09:00.600
Well, sure.

00:09:00.600 --> 00:09:04.130
This is in our distance equation
that we did before,

00:09:04.130 --> 00:09:10.500
where we just say that that
change in distance is equal to

00:09:10.500 --> 00:09:17.460
the initial velocity times time
plus at squared over 2.

00:09:21.170 --> 00:09:26.780
This term right here looks an
awfully lot like this term,

00:09:26.780 --> 00:09:30.170
and this would be equivalent
to this equation if this

00:09:30.170 --> 00:09:32.670
initial velocity was 0.

00:09:32.670 --> 00:09:36.090
So how does this relate?

00:09:36.090 --> 00:09:40.440
Another way to think about it
is this distance that you're

00:09:40.440 --> 00:09:48.990
traveling up, it's also the same
distance-- maybe just the

00:09:48.990 --> 00:09:52.045
negative-- of the distance
that you travel down.

00:09:56.520 --> 00:10:02.520
If we said that we're starting
at this point, at this point

00:10:02.520 --> 00:10:05.540
the ball is stationary, so it's
identical to just taking

00:10:05.540 --> 00:10:09.660
that ball, and dropping it.

00:10:09.660 --> 00:10:13.300
If we say that we're going to
drop a ball from height h, how

00:10:13.300 --> 00:10:15.000
long does it take?

00:10:15.000 --> 00:10:17.240
Then we can use this formula.

00:10:17.240 --> 00:10:21.200
If I drop a ball from height h,
the change in distance is

00:10:21.200 --> 00:10:24.710
going to be minus h, because
it's going to go from height h

00:10:24.710 --> 00:10:29.460
to 0, and so we could say that
minus h is equal to-- the

00:10:29.460 --> 00:10:32.930
initial velocity is
going to be 0.

00:10:32.930 --> 00:10:33.260
Whoops!

00:10:33.260 --> 00:10:34.860
I'm over time.


Projectile motion (part 5)

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:01.920
Welcome back.

00:00:01.920 --> 00:00:05.180
Let's continue doing projectile
motion problems. I

00:00:05.180 --> 00:00:07.380
think this video will be
especially entertaining,

00:00:07.380 --> 00:00:09.370
because I will teach you a game
that you can play with a

00:00:09.370 --> 00:00:13.370
friend, and it's called let's
see how fast and how high I

00:00:13.370 --> 00:00:15.110
can throw this ball.

00:00:15.110 --> 00:00:18.270
You'd be surprised--
it's actually quite

00:00:18.270 --> 00:00:20.020
a stimulating game.

00:00:20.020 --> 00:00:22.910
Let me just write
down everything

00:00:22.910 --> 00:00:24.100
we've learned so far.

00:00:24.100 --> 00:00:32.119
Change in distance is
equal to the average

00:00:32.119 --> 00:00:36.540
velocity times time.

00:00:36.540 --> 00:00:43.170
We know that change in
velocity is equal to

00:00:43.170 --> 00:00:45.340
acceleration times time.

00:00:45.340 --> 00:00:48.600
We also know that average
velocity is equal to the final

00:00:48.600 --> 00:00:53.800
velocity plus the initial
velocity over 2.

00:00:53.800 --> 00:00:56.400
We know the change in velocity,
of course, is equal

00:00:56.400 --> 00:00:58.960
to the final velocity minus
the initial velocity.

00:00:58.960 --> 00:01:00.775
This should hopefully be
intuitive to you, because it's

00:01:00.775 --> 00:01:02.660
just how fast you're going at
the end, minus how fast you're

00:01:02.660 --> 00:01:05.080
going at the beginning,
divided-- oh no, no division,

00:01:05.080 --> 00:01:08.080
it's just that I got
stuck in a pattern.

00:01:08.080 --> 00:01:09.900
It's just vf minus
vi, of course.

00:01:14.500 --> 00:01:17.560
You probably already knew this
before you even stumbled upon

00:01:17.560 --> 00:01:21.340
my videos, but-- the two
non-intuitive ones that we've

00:01:21.340 --> 00:01:23.980
learned, they're really just
derived from what I've just

00:01:23.980 --> 00:01:24.470
written up here.

00:01:24.470 --> 00:01:26.120
If you ever forget them, you
should try to derive them.

00:01:26.120 --> 00:01:28.130
Actually, you should try to
derive them, even if you don't

00:01:28.130 --> 00:01:29.780
forget them, so that you
when you do forget it,

00:01:29.780 --> 00:01:31.000
you can derive it.

00:01:31.000 --> 00:01:37.020
It's change in distance-- let
me change it to lowercase d,

00:01:37.020 --> 00:01:42.680
just to confuse you-- is equal
to the initial velocity times

00:01:42.680 --> 00:01:52.760
time plus at squared over 2,
and that's one of what I'll

00:01:52.760 --> 00:01:55.110
call the non-intuitive
formulas.

00:01:55.110 --> 00:01:59.600
The other one is the final
velocity squared is equal to

00:01:59.600 --> 00:02:03.200
the initial velocity
squared plus 2ad.

00:02:03.200 --> 00:02:05.090
We've derived all of these,
and I encourage you to try

00:02:05.090 --> 00:02:07.060
rederive them.

00:02:07.060 --> 00:02:12.070
But using these two formulas
you can play my fun game of

00:02:12.070 --> 00:02:14.630
how fast and how high did
I throw this ball?

00:02:14.630 --> 00:02:19.890
All you need is your arm, a
ball, a stopwatch, and maybe

00:02:19.890 --> 00:02:23.200
some friends to watch
you throw the ball.

00:02:23.200 --> 00:02:24.460
So how do we play this game?

00:02:24.460 --> 00:02:28.110
We take a ball, and we throw
it as high as we can.

00:02:28.110 --> 00:02:31.760
We see how long does the
ball stay in the air?

00:02:31.760 --> 00:02:32.810
What do we know?

00:02:32.810 --> 00:02:37.120
We know the time for the ball
to leave your hand, to

00:02:37.120 --> 00:02:39.970
essentially leave the ground and
come back to the ground.

00:02:39.970 --> 00:02:48.240
We are given time, and
what else do we know?

00:02:48.240 --> 00:02:51.570
We know acceleration-- we know
acceleration is this minus 10

00:02:51.570 --> 00:02:52.670
meters per second.

00:02:52.670 --> 00:02:55.130
If you're actually playing
this game for money, or

00:02:55.130 --> 00:02:57.770
something, you would probably
want to use a more accurate

00:02:57.770 --> 00:03:00.550
acceleration-- you could look
it up on Wikipedia.

00:03:00.550 --> 00:03:07.400
It's minus 9.81 meters
per second squared.

00:03:07.400 --> 00:03:09.680
Do we know the change
in distance?

00:03:09.680 --> 00:03:12.060
At first, you're saying-- Sal,
I don't know how high this

00:03:12.060 --> 00:03:14.540
ball went, but we're talking
about the change in distance

00:03:14.540 --> 00:03:17.700
over the entire time.

00:03:17.700 --> 00:03:19.430
It starts at the ground--
essentially at the ground,

00:03:19.430 --> 00:03:21.266
because I'm assuming that you're
not 100 feet tall, and

00:03:21.266 --> 00:03:24.050
so you're essentially at the
ground-- so it starts at the

00:03:24.050 --> 00:03:26.610
ground, and it ends of the
ground, so the change in

00:03:26.610 --> 00:03:33.260
distance of delta d is 0.

00:03:33.260 --> 00:03:35.180
It starts with at the ground
and ends at the ground.

00:03:35.180 --> 00:03:38.080
This is interesting-- this is a
vector quantity, because the

00:03:38.080 --> 00:03:38.880
direction matters.

00:03:38.880 --> 00:03:41.460
If I told you how far did the
ball travel, then you'd have

00:03:41.460 --> 00:03:43.720
to look at its path, and say
how high did it go, and how

00:03:43.720 --> 00:03:45.690
high would it come back?

00:03:45.690 --> 00:03:48.000
Actually, if you want to be
really exact, the change in

00:03:48.000 --> 00:03:50.930
distance would be the height
from your hand when the ball

00:03:50.930 --> 00:03:54.880
left your hand, to the ground--
so, if you're 6 feet

00:03:54.880 --> 00:03:57.440
tall, or 2 meters tall, the
change in distance would

00:03:57.440 --> 00:03:59.490
actually be minus 2 meters,
but we're not going to do

00:03:59.490 --> 00:04:02.870
that, because that would just be
too much, but you could do

00:04:02.870 --> 00:04:06.190
it if there's ever a close call
between you and a friend,

00:04:06.190 --> 00:04:08.280
and you're betting for money.

00:04:08.280 --> 00:04:10.830
You're given these things,
and we want to figure

00:04:10.830 --> 00:04:13.300
out a couple of things.

00:04:13.300 --> 00:04:16.410
The first thing I want to figure
out is how fast did I

00:04:16.410 --> 00:04:19.560
throw the ball, because that's
what's interesting-- that

00:04:19.560 --> 00:04:22.800
would be a pure test
of testosterone.

00:04:22.800 --> 00:04:23.450
How fast?

00:04:23.450 --> 00:04:30.470
I want to figure out vi--
vi equals question mark.

00:04:30.470 --> 00:04:31.640
Which of these formulas
can be used?

00:04:31.640 --> 00:04:33.300
Actually, I'm going do it first
with the formulas, and

00:04:33.300 --> 00:04:35.750
then I'm going to show you
almost an easier way to do it,

00:04:35.750 --> 00:04:36.640
where it's more intuitive.

00:04:36.640 --> 00:04:40.440
I want to show you that these
formulas can be used for fun

00:04:40.440 --> 00:04:42.310
with your friends.

00:04:42.310 --> 00:04:46.060
We know time, we know
acceleration, we know the

00:04:46.060 --> 00:04:49.930
change in distance, so we could
just solve for vi--

00:04:49.930 --> 00:04:51.350
let's do that.

00:04:51.350 --> 00:04:55.310
In this situation, change in
distance is 0-- let me change

00:04:55.310 --> 00:05:00.730
colors again just to change
colors-- so change in distance

00:05:00.730 --> 00:05:09.450
is 0 is equal to
vi times time.

00:05:09.450 --> 00:05:13.420
Let me put the g in for here,
so it's minus 10 meters per

00:05:13.420 --> 00:05:17.850
second squared divided by 2,
and it's minus 5 meters per

00:05:17.850 --> 00:05:24.720
second squared-- so it's
minus 5t squared.

00:05:24.720 --> 00:05:27.170
All I did it is that I took
minus 10 meters per second

00:05:27.170 --> 00:05:29.400
squared for a, divided it
by 2, and that's how I

00:05:29.400 --> 00:05:30.070
got the minus 5.

00:05:30.070 --> 00:05:32.510
If you used 9.81 or whatever,
this would be

00:05:32.510 --> 00:05:35.610
4.905 something something.

00:05:35.610 --> 00:05:37.630
Anyway, let's get back
to the problem.

00:05:37.630 --> 00:05:42.000
If we wanted to solve this
equation for vi,

00:05:42.000 --> 00:05:44.860
what could we do?

00:05:44.860 --> 00:05:46.900
This is pretty interesting,
because we

00:05:46.900 --> 00:05:49.910
could factor a t out.

00:05:49.910 --> 00:05:51.700
What's cool about these physics
equations is that

00:05:51.700 --> 00:05:55.190
everything we do actually has
kind of a real reasoning

00:05:55.190 --> 00:05:57.590
behind it in the real world,
so let me actually flip the

00:05:57.590 --> 00:06:01.600
sides, and factor out a t, just
to make it confusing.

00:06:01.600 --> 00:06:12.210
I get t times vi minus
5t is equal to 0.

00:06:12.210 --> 00:06:14.770
All I did is that I factored out
a t, and I could do this--

00:06:14.770 --> 00:06:16.710
I didn't have to use a quadratic
equation, or factor,

00:06:16.710 --> 00:06:18.930
because there wasn't a
constant term here.

00:06:18.930 --> 00:06:22.320
So I have this expression, and
if I were to solve it,

00:06:22.320 --> 00:06:25.110
assuming that you know vi is
some positive number, I know

00:06:25.110 --> 00:06:29.160
that there's two times where
this equation is true.

00:06:29.160 --> 00:06:41.810
Either t equals 0, or this term
equals 0-- vi minus 5t is

00:06:41.810 --> 00:06:48.460
equal to 0, or since I'm solving
for velocity, we know

00:06:48.460 --> 00:06:54.323
that vi is equal to 5t.

00:06:57.198 --> 00:06:58.880
That's interesting.

00:06:58.880 --> 00:07:01.070
So what does this say?

00:07:01.070 --> 00:07:03.050
If we knew the velocity, we
could solve it the other way.

00:07:03.050 --> 00:07:08.580
We could say that t is equal to
vi divided by 5-- these are

00:07:08.580 --> 00:07:11.370
the same things, just solving
for a different variable.

00:07:11.370 --> 00:07:16.790
But that's cool, because there
are two times when the change

00:07:16.790 --> 00:07:20.000
in distance is zero-- at time
equals 0, of course, the

00:07:20.000 --> 00:07:22.430
change in distance is zero,
because I haven't thrown the

00:07:22.430 --> 00:07:27.150
ball yet, and then, at a later
time, or my initial velocity

00:07:27.150 --> 00:07:29.530
divided by 5, it'll also
hit the ground again.

00:07:29.530 --> 00:07:30.930
Those are the two times
that the change

00:07:30.930 --> 00:07:31.740
in distance is zero.

00:07:31.740 --> 00:07:32.330
That's pretty cool.

00:07:32.330 --> 00:07:34.810
This isn't just math--
everything we're doing in math

00:07:34.810 --> 00:07:38.550
has kind of an application
in the real world.

00:07:38.550 --> 00:07:42.570
We've solved our equation--
vi is equal to 5t.

00:07:42.570 --> 00:07:49.150
So, if you and a friend go
outside and throw a ball, and

00:07:49.150 --> 00:07:50.710
you try to throw it straight
up--0 although we'll learn

00:07:50.710 --> 00:07:53.540
when we do the two dimensional
projectile motion that it

00:07:53.540 --> 00:07:56.000
actually doesn't matter if you
have a little bit of an angle

00:07:56.000 --> 00:07:59.340
on it, because the vertical
motion and the horizontal

00:07:59.340 --> 00:08:01.390
motions are actually
independent, or can be viewed

00:08:01.390 --> 00:08:06.070
as independent from each other--
this velocity you're

00:08:06.070 --> 00:08:09.160
going to get if you play this
game is going to be just the

00:08:09.160 --> 00:08:12.670
component of your velocity
that goes straight up.

00:08:12.670 --> 00:08:14.645
I know that might be a little
confusing, and hopefully it

00:08:14.645 --> 00:08:16.270
will make a little more sense in
a couple of videos from now

00:08:16.270 --> 00:08:17.520
when I teach you vectors.

00:08:20.030 --> 00:08:24.940
If you were to throw this ball
straight up, and time when it

00:08:24.940 --> 00:08:27.120
hits the ground, then this
velocity would literally the

00:08:27.120 --> 00:08:28.990
speed-- actually, the
velocity-- at which

00:08:28.990 --> 00:08:30.230
you throw the ball.

00:08:30.230 --> 00:08:31.410
So what would it be?

00:08:31.410 --> 00:08:39.950
If I threw a ball, and it took
two seconds to go up hit the

00:08:39.950 --> 00:08:42.400
ground, then I could
use this formula.

00:08:42.400 --> 00:08:44.890
This is actually 5
meters per second

00:08:44.890 --> 00:08:51.490
squared times t seconds.

00:08:51.490 --> 00:08:57.710
If it took 2 seconds-- so if
t is equal to 2-- then my

00:08:57.710 --> 00:09:01.100
initial velocity is equal
to 10 meters per second.

00:09:01.100 --> 00:09:03.160
You could convert that to
miles per hour-- we've

00:09:03.160 --> 00:09:05.980
actually done that in
previous videos.

00:09:05.980 --> 00:09:16.380
If you throw a ball that stays
up in the air for 10 seconds,

00:09:16.380 --> 00:09:19.780
then you threw it at 50 meters
per second, which is

00:09:19.780 --> 00:09:24.170
extremely, extremely fast.
Hopefully, I've taught you a

00:09:24.170 --> 00:09:25.560
little bit about a fun game.

00:09:25.560 --> 00:09:28.040
In the next video, I'll show
you how to figure out-- how

00:09:28.040 --> 00:09:29.380
high did the ball go?

00:09:29.380 --> 00:09:30.790
I'll see you soon.


Projectile motion (part 4)

WEBVTT
Kind: captions
Language: en

00:00:01.060 --> 00:00:05.500
We'll now use that equation we
just derived to go back and

00:00:05.500 --> 00:00:07.980
solve-- or at least address--
that same problem we were

00:00:07.980 --> 00:00:10.510
doing before, so let's write
that equation down again.

00:00:10.510 --> 00:00:11.660
Actually, let's write
the problem down.

00:00:11.660 --> 00:00:15.140
Lets say I have the cliff
again, and so my initial

00:00:15.140 --> 00:00:18.750
distance is 0, but it goes
down 500 meters.

00:00:18.750 --> 00:00:20.230
I'm not going to redraw the
cliff, because it takes a lot

00:00:20.230 --> 00:00:23.260
of space up on my limited
chalkboard.

00:00:23.260 --> 00:00:27.440
We know that the change
in distance is equal

00:00:27.440 --> 00:00:32.659
to minus 500 meters.

00:00:32.659 --> 00:00:35.100
I'm still going to use the
example where I don't just

00:00:35.100 --> 00:00:38.070
drop the ball, or the penny, or
whatever I'm throwing off

00:00:38.070 --> 00:00:40.270
the cliff, but I actually throw
it straight up, so it's

00:00:40.270 --> 00:00:43.145
going to go up and slow down
from gravity, and then it will

00:00:43.145 --> 00:00:46.370
go to 0 velocity, and start
accelerating downwards.

00:00:46.370 --> 00:00:47.275
You could even say decelerating

00:00:47.275 --> 00:00:48.930
in the other direction.

00:00:48.930 --> 00:00:59.780
The initial velocity, vi, is
equal to 30 meters per second,

00:00:59.780 --> 00:01:02.730
and of course, we know that the
acceleration is equal to

00:01:02.730 --> 00:01:06.530
minus 10 meters per second
squared-- it's because

00:01:06.530 --> 00:01:09.200
acceleration gravity is always
pulling downwards, or towards

00:01:09.200 --> 00:01:11.260
the center of our planet.

00:01:11.260 --> 00:01:13.150
If we wanted to figure out the
final velocity, we could have

00:01:13.150 --> 00:01:15.150
just used the formula, and we
did this in the last video--

00:01:15.150 --> 00:01:21.340
vf squared is equal to
vi squared plus 2ad.

00:01:21.340 --> 00:01:24.900
Now what I want to do is use the
formula that we learned in

00:01:24.900 --> 00:01:27.260
the very last video to figure
out-- how long does it take to

00:01:27.260 --> 00:01:30.680
get to the bottom, and
to hit the ground?

00:01:30.680 --> 00:01:37.350
Let's use that formula: we
derived that the change in

00:01:37.350 --> 00:01:43.720
distance is equal to the initial
velocity times time

00:01:43.720 --> 00:01:48.570
plus acceleration time
squared over 2, and

00:01:48.570 --> 00:01:50.200
that's initial velocity.

00:01:50.200 --> 00:01:55.550
The change in distance is minus
500, and that's equal to

00:01:55.550 --> 00:01:59.410
the initial velocity-- that's
positive, going upwards, 30

00:01:59.410 --> 00:02:02.300
meters per second, 30t.

00:02:02.300 --> 00:02:04.600
I'm not going to write the units
right now, because I'll

00:02:04.600 --> 00:02:07.030
run out of space, but you can
redo it with the units, and

00:02:07.030 --> 00:02:08.550
see that the units
do work out.

00:02:08.550 --> 00:02:11.009
When you square time, you have
to square the time units,

00:02:11.009 --> 00:02:12.792
although we're solving
for time.

00:02:12.792 --> 00:02:18.250
Then, plus acceleration, and
acceleration is minus 10, and

00:02:18.250 --> 00:02:25.850
we're going to divided it by 2,
so it's minus 5t squared.

00:02:25.850 --> 00:02:30.520
We have minus 500 is equal to
30t plus minus 5t, and we

00:02:30.520 --> 00:02:32.000
could just say minus
5t squared, and

00:02:32.000 --> 00:02:34.270
get rid of this plus.

00:02:34.270 --> 00:02:37.150
At first, you say, Sal-- there's
a t, that's t to the

00:02:37.150 --> 00:02:39.600
first, and t to the second,
how do I saw solve this?

00:02:39.600 --> 00:02:43.310
Hopefully, you've taken algebra
two or algebra one, in

00:02:43.310 --> 00:02:45.430
some places, and you remember
how to solve this.

00:02:45.430 --> 00:02:47.310
Otherwise, you're about to learn
the quadratic equation,

00:02:47.310 --> 00:02:49.150
although I recommend you go
back, and learn about

00:02:49.150 --> 00:02:51.430
factoring in the quadratic
equation, which there are

00:02:51.430 --> 00:02:54.280
videos on that I've
put on Youtube.

00:02:54.280 --> 00:02:56.570
I hope you watch those first
if you don't remember.

00:02:56.570 --> 00:02:59.080
We can do this-- let's put these
two right terms on the

00:02:59.080 --> 00:03:01.810
left hand side, and then we'll
use the quadratic equation to

00:03:01.810 --> 00:03:03.625
solve, because I don't think
this is easy to factor.

00:03:03.625 --> 00:03:14.020
We'll get 5t squared minus 30t
minus 500 is equal to 0-- I

00:03:14.020 --> 00:03:17.060
just took these terms and put
them on the left side.

00:03:17.060 --> 00:03:20.580
We could divide both sides by
5, just to simplify things,

00:03:20.580 --> 00:03:27.740
and so we get t squared minus
6t minus 100 is equal to 0.

00:03:27.740 --> 00:03:30.160
I could do that, because 0
divided by 5 is just five, so

00:03:30.160 --> 00:03:32.220
I just cleaned it
up a little bit.

00:03:32.220 --> 00:03:35.130
Let's use the quadratic
equation, and for those of us

00:03:35.130 --> 00:03:37.840
who need a refresher,
I'll write it down.

00:03:37.840 --> 00:03:43.740
The roots of any quadratic--
in this case, it's t we're

00:03:43.740 --> 00:03:48.060
solving for-- t will equal
negative b plus or minus the

00:03:48.060 --> 00:03:56.770
square root of b squared minus
4ac over 2a, where a is a

00:03:56.770 --> 00:03:59.990
coefficient on this term, b is
the coefficient on this term,

00:03:59.990 --> 00:04:03.900
negative 6, and c is the
constant, so minus 100.

00:04:03.900 --> 00:04:05.830
Let's just solve.

00:04:05.830 --> 00:04:11.190
We get t is equal to negative
b-- so negative this term.

00:04:11.190 --> 00:04:14.850
This term is negative 6, so if
we make it a negative, it

00:04:14.850 --> 00:04:16.380
becomes plus 6.

00:04:16.380 --> 00:04:23.305
It becomes 6 plus or minus the
square root of b squared, so

00:04:23.305 --> 00:04:33.130
it's minus 6 squared, 36,
minus 4 times a, and the

00:04:33.130 --> 00:04:36.826
coefficient on a is here,
and that's just times 1.

00:04:36.826 --> 00:04:42.810
With 4ac, c is a constant term,
minus 100, minus 4 times

00:04:42.810 --> 00:04:51.250
1 times minus 100, and all of
that is over 2a-- a is 1

00:04:51.250 --> 00:04:54.010
agains, so all of
that is over 2.

00:04:54.010 --> 00:04:59.140
That just equals 6 plus or minus
the square root-- this

00:04:59.140 --> 00:05:02.470
is minus 4 times minus 100, and
these become pluses, so it

00:05:02.470 --> 00:05:05.490
becomes 36 plus 400.

00:05:05.490 --> 00:05:15.180
So, 6 plus or minus
436 divided by 2.

00:05:15.180 --> 00:05:17.680
This is not a clean number,
and if you type into a

00:05:17.680 --> 00:05:22.050
calculator, it's something on
the order of about 20.9.

00:05:22.050 --> 00:05:24.730
We can just say approximately
21-- you might want to get the

00:05:24.730 --> 00:05:26.710
exact number, if you're actually
doing this on a test,

00:05:26.710 --> 00:05:30.880
or trying to send something to
Mars, but for our purposes, I

00:05:30.880 --> 00:05:32.420
think you'll get the point.

00:05:32.420 --> 00:05:34.080
I'll say it approximately now,
because we're going to be a

00:05:34.080 --> 00:05:36.210
little off, but just to have
clean numbers, this is

00:05:36.210 --> 00:05:39.540
approximately 21--
it's like 20.9.

00:05:39.540 --> 00:05:43.260
We'll say 6 plus or minus--
let me just write

00:05:43.260 --> 00:05:52.410
20.9-- 20.9 over 2.

00:05:52.410 --> 00:05:58.250
Let me ask you a question:
if I do 6 minus

00:05:58.250 --> 00:06:00.940
20.9, what do I get?

00:06:00.940 --> 00:06:02.770
I get a negative number,
and does a

00:06:02.770 --> 00:06:05.340
negative time make sense?

00:06:05.340 --> 00:06:08.120
No, it does not, and that means
that somehow in the

00:06:08.120 --> 00:06:11.100
past-- I don't want to get
philosophical-- the negative

00:06:11.100 --> 00:06:13.360
time in this context will
not make sense.

00:06:13.360 --> 00:06:16.342
Really, we can just stick to the
plus, because 6 minus 20

00:06:16.342 --> 00:06:20.540
is negative, so there's only one
time that will solve this

00:06:20.540 --> 00:06:22.570
in a meaningful way.

00:06:22.570 --> 00:06:28.900
Time is approximately equal to
6 plus 20.9, so that's 26.9

00:06:28.900 --> 00:06:39.690
over 2, and that equals
13.45 seconds.

00:06:39.690 --> 00:06:40.670
That's interesting.

00:06:40.670 --> 00:06:43.490
I think if you remember way
back, maybe four or five

00:06:43.490 --> 00:06:47.110
videos ago, when we first did
this problem, we just dropped

00:06:47.110 --> 00:06:49.530
the penny straight
from the height.

00:06:49.530 --> 00:06:52.095
Actually, in that problem, I
gave you the time-- I said it

00:06:52.095 --> 00:06:54.790
took 10 seconds to hit the
ground, and we worked

00:06:54.790 --> 00:06:58.660
backwards to figure out that the
cliff was 500 meters high.

00:06:58.660 --> 00:07:02.660
Now, if you're here at the top
of a 500 cliff, or building,

00:07:02.660 --> 00:07:04.970
and you drop something that has
air resistance-- like a

00:07:04.970 --> 00:07:07.700
penny, that has very air
resistance-- it would take 10

00:07:07.700 --> 00:07:10.060
seconds to reach the ground,
assuming all of our

00:07:10.060 --> 00:07:12.000
assumptions about gravity.

00:07:12.000 --> 00:07:15.280
But if you were to throw the
penny straight up, off the

00:07:15.280 --> 00:07:19.470
edge of the cliff, at 30 meters
per second right here,

00:07:19.470 --> 00:07:23.720
it's going to take 13.5--
roughly, 13 and 1/2 seconds--

00:07:23.720 --> 00:07:24.680
to reach the ground.

00:07:24.680 --> 00:07:27.460
It takes a little bit longer,
and that should make sense

00:07:27.460 --> 00:07:33.230
because-- I have time to
draw a little picture.

00:07:33.230 --> 00:07:43.040
In the first case, I just took
the penny, and its motion just

00:07:43.040 --> 00:07:45.220
went straight down.

00:07:45.220 --> 00:07:51.310
In the second case, I took the
penny-- it first went up, and

00:07:51.310 --> 00:07:52.930
then it went down.

00:07:55.440 --> 00:07:59.600
It had all the time when it went
up, and then it went down

00:07:59.600 --> 00:08:03.310
a longer distance, so it makes
sense that this time-- this

00:08:03.310 --> 00:08:09.950
was 10 seconds, while this time
was 13.45 five seconds.

00:08:09.950 --> 00:08:13.310
You can kind of say that it
took-- well, you actually

00:08:13.310 --> 00:08:15.920
can't say that.

00:08:15.920 --> 00:08:20.000
I don't want to get too
involved, but I hopefully this

00:08:20.000 --> 00:08:20.710
make sense to you.

00:08:20.710 --> 00:08:23.620
If you have a smaller number
here, you should have gone and

00:08:23.620 --> 00:08:25.600
checked your work, because why
would it take less time when I

00:08:25.600 --> 00:08:30.030
throw the object straight up?

00:08:30.030 --> 00:08:33.370
Hopefully, that gave you a
little bit more intuition, and

00:08:33.370 --> 00:08:36.620
you really do have in your
arsenal now all of the

00:08:36.620 --> 00:08:40.049
equations-- and hopefully, the
intuition you need-- to solve

00:08:40.049 --> 00:08:41.480
basic projectile problems.

00:08:41.480 --> 00:08:43.960
I'll now probably do a couple
more videos where I just do a

00:08:43.960 --> 00:08:47.580
bunch of problems, just to
really drive the points home.

00:08:47.580 --> 00:08:50.570
Then, I'll expand these
problems to two

00:08:50.570 --> 00:08:52.700
dimensions and angles.

00:08:52.700 --> 00:08:54.510
Before we get there, you might
want to refresh your

00:08:54.510 --> 00:08:55.440
trigonometry.

00:08:55.440 --> 00:08:57.060
I'll see you soon.