in this example, we are given that water stands
upto a height h in a tank, as shown here.
and a hole is made at a depth small h, below
the open end of the tank, from which water
ejects out. we are required to find the value
of this h, from which the range of ejecting
water on ground will be maximum. so here we
need to maximize this range x. say if the
hole is made at, a height y above the ground
level, then y can be given as, capital h minus
small h. in this situation efflux velocity
by torecellie’s theorem we already know.
here, as we know, the efflux velocity, of
water is, this v can be written as, root 2
g h. and if we find out the time taken by
the water particles which are coming out to
reach the ground, as they are in free fall
state, for a height h minus h, so we can find
out, time to reach, ground level, by water
is, this time can be written as, root 2 y
by g, which can be written as, root of twice,
h minus, small h by g. and, throughout motion
as g is acting in downward direction we know,
the speed of, water ejected out in horizontal
direction remains same. so we can simply calculate,
the horizontal range on ground is, x can be
written as, v t. this can be directly written
as, root 2 g h, multiplied by, root of twice
of, h minus small h by g, here g gets cancelled
out. the result will be twice of, root small
h into, h minus small h. now we can simply
state, for x to be maximum, we need to use,
d x by, d h, equal to zero. and this implies,
if we differentiate the expression, twice
of root h into h minus h, it can be given
as, 1 by twice of, root of h, into h minus
small h, multiplied by the differentiation
of this term, which can be written as, h minus,
2 small h, this should be equal to zero. 2
also is their i have not written because it
is a constant. on simplifying we can get the
value of small h to be equal to, h by 2. that’ll
be the answer to this problem. and h is equal
to, capital h by 2, this is for d x by d h
should be zero or for x to be, maximum in
this situation.
