## Friday, 22 June 2018

### Exercise 1.01 Bouncing Ball

I have read and understood most of chapter 1. I am slightly stuck on the final section about Classical Field Theory because I have forgotten or was never taught much about this at university. So I decided to do a few of the exercises. The first one was Exercise 1.01.

## Question

Consider an inertial frame S with coordinates   xμ = ( t , x , y , z )  and a frame S' with coordinates xμ' related to S by a boost with velocity parameter v along the y-axis. Imagine we have a wall at rest in S', lying along the line x' = -y'. From the point of view of S, what is the relationship between the incident angle of a ball hitting the wall (travelling in the x-y plane) and the reflected angle? What about the velocity before and after?

I found this the most difficult of the first seven exercises and gave up and came back to it. I think I have the right answer now. Sometimes the reflected ball has an angle of reflection > 90°, which means it seems to go through the wall, as illustrated below.
In fact, the wall is moving upwards so fast that it stays ahead of the ball.

$$\large \frac{1}{c^2}\frac{{\gamma }_{\mathrm{v}}}{{\gamma }_{\mathrm{v}}+1}=\frac{1}{v^2}\left(1-\frac{1}{{\gamma }_{\mathrm{v}}}\right)$$which I proved in the complete answer above and here.