solution

Let's start by drawing some pictures

At first, the free mass has a velocity $v$ and does a one dimensional elastic collision with the first mass. As we know, when equal masses have elastic collisions, they exchange velocities.

(a) In other words, after the collision, the free mass is at rest, and the mass on the spring starts moving with a velocity of $v$.

(b) We can use conservation of energy. The initial energy right after the collision, of the mass on the spring is ${1 \over 2}mv^2$. At maximum compression all the energy is potential, so

$${1 \over 2}mv^2 ~=~ {1 \over 2}kx^2$$ (1.13)

Therefore the maximum compression is

$$x ~=~ \sqrt{m\over k}v$$ (1.14)

(c) From the picture, you can see that the time interval is half a period. But $\omega ~=~ {2\pi\over T}$ so

$$T ~=~ {2\pi\over \omega} ~=~ 2\pi\sqrt{m\over k}$$ (1.15)

So the time interval is $T/2 ~=~ \pi\sqrt{m\over k}$. Note that this is independent of the initial velocity.

(d) When the first mass recollides with the second, they exchange velocities again. So you end up with the mass on the spring at rest, and the free mass flying off with velocity $-v$, the negative of the velocity it initially had before colliding with the mass on the spring.