Newtonian-Mechanics .Elastic reflection Question from the olympiad.

Two planes move perpendicular to each other with speeds $v = 1m/s$. A body flies at them with a velocity of $u = 1 m/s$ at an angle $a=45$ to the surfaces of the planes and is elastically reflected successively from each of the planes.No friction
Question: Find approximate expression for: 
  
  
*
  
*Final velocity of the body if its movement occurs in a plane perpendicular to the original planes
If the mass of body not indicated


My attempt at a solution: There should have been my attempts to solve the problem, but I have no ideas about this. I have a more fundamental question. How to understand what happens to the body when it is reflected elastically. I was told that the momentum and kinetic energy do not change when the reflection is elastic. Then please using this problem as an example, show how to use this fact. I first encountered a similar problem and therefore please explain the physics of this problem. And how I could solve this problem myself. 
Thank you in advance for the answer.
 A: Note that initially I thought that it might be appropriate to use conservation of momentum of the system for this question but the problem was that masses of the objects aren't given. But then John Rennie sir hinted that it might be the case that the planes (on which the ball is going to hit) have infinite mass. This makes the problem simpler (if that is the case).

Since the planes have infinite mass therefore they just act as a rigid wall and reflect the ball with whatever the speed it came with, just you have to make sure that the speed that you account for is the one relative to the wall i.e., to say you find the relative velocity of the ball wrt wall
$$\mathbf v_{\text {ball,wall}} = \mathbf v_{\text {ball}} - \mathbf v_{\text {wall}}$$
Now do the same with the collision at the second wall.
A: Assuming that the wall acts like a very large mass and the collision is elastic, then the rebound speed and angle are the same before and after the collision when  measured relative to the center of mass (the wall).  Taking x to the right and y up on the sketch, the initial relative x component for the first collision is ux + v and the final is -(ux + v).    In the fixed system, the final x component is -(ux + 2v).  Assuming no friction with the wall, the y component does not change.  For the second collision, the logic is similar but rotated by 90 degrees. 
