Asteroids X, Y, and Z have equal mass of 5.0 kg each. They orbit around a planet

Question

Asteroids X, Y, and Z have equal mass of 5.0 kg each. They orbit around a planet with M=4.20E+24 kg. The orbits are in the plane of the paper and are drawn to scale

In the statements below, TE is the total mechanical energy, KE is the kinetic energy, and PE is the potential energy.  

greater than less than equal to  The PE of X at r is .... the PE of Y at u
greater than less than equal to  The PE of Z at s is .... that at m
greater than less than equal to  The speed of X at m is .... that at c
greater than less than equal to  The KE of Y at r is .... that at u
greater than less than equal to  The TE of Z is .... the TE of Y
greater than less than equal to  The PE of Z at u is .... the PE of Y at u
greater than less than equal to  The TE of X is .... the TE of Z
greater than less than equal to  The PE of X at m is .... the PE of Z at u

Explanation / Answer

You need the diagram to solve the problem as distances are not known but still I have given the concept.

We can think of X and Y as projectiles launched from the orbit of Z. The source gives another way of seeing the problem : the specific orbital energy is inversely proportional to the major axis of the orbit.

i. The PE of X at u is GREATER THAN the PE of Y at r
(PE increases with distance [=height] from attractive body.) (important point)

ii. The PE of Y at u is THE SAME AS the PE of X at u (since the asteroids have the same mass)

iii. The PE of X at s is LESS that at m (see i)

iv. The TE of X is GREATER THAN the TE of Y (TE is constant; X reaches a greater "height")

v. The speed of X at u is GREATER THAN that at i (height is less; speed decreases with height)

vi. The PE of X at s is THE SAME AS the PE of Z at r (same mass, same distance from planet)

vii. The TE of Z is LESS THAN the TE of Y (Y reaches a greater height from the planet)

viii. The KE of Z at s is THE SAME AS that at r (constant TE, same PE, therefore same KE)

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