A comet orbits the Sun in the same plane as Earth's orbit If the Earth around the Sun follows a practically circular orbit of radius equal to one astronomical unit, determine its orbital speed in astronomical units per year.
If the Earth around the Sun follows a practically circular orbit of radius equal to one astronomical unit, determine its orbital speed in astronomical units per year.
A comet orbits the Sun in the same plane as Earth's orbit. The comet's closest approach to the center of the Sun is 0.5 u.a. and it has, at that point, an orbital velocity of 4pi u.a / year. What is the orientation of the comet at the moment it cuts the Earth's orbit
I'll just give an outline about how I might try to answer these questions, rather than answering them outright.
Answer the first question using the circular form of the Vis-Viva equation: $v=\sqrt{\frac{\mu}{r}}$, where $\mu$ is the standard gravitational parameter ($4\pi^2 AU^3/yr^2$) and $r$ is 1 AU.  If your answer isn't $2\pi$ AU per year, you probably did something wrong with the units.
For the second question, try to determine the shape of the comet's orbit using the inequality for escape velocity: $v_e \ge \sqrt{\frac{2\mu}{r}}$.  Once you have the orbit shape, the rest of the problem falls out pretty easily using known equations for that orbit.

 Escape velocity at 0.5 AU is
 $$v_e=\sqrt{\frac{2*4\pi^2}{0.5}}=\sqrt{16\pi^2}=4\pi$$

 which is the exact speed given in the question. That means the orbit
 is parabolic.  From  Bogan's table pulled from
 Bate/Mueller/White's Fundamentals of Astrodynamics, for a parabolic
 orbit, $$r=\frac{2q}{1+\cos{\theta}}$$ where $r$ is the distance from
 the central body, $q$ is the closest distance of the orbit, and
 $\theta$ is the angle from periapsis or true anomaly. Solving gives us
 $\theta=\pi/2$.

 The same table gives us the angle of velocity relative to the
 perpendicular to the radial direction: $$\phi = \theta/2 = \pi/4$$

 So the orientation of the comet's velocity is at
 $\pi/4$ or 45 degrees off a tangent line to a circle around the Sun
 with radius 1 AU as the comet passes through it.