## 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.

In order to check my answer I used an equation on Wikipedia for relativistic velocity transformations here. But I first had to slightly improve it by showing that
$$\large \frac{1}{c^2}\frac{{\gamma }_{\mathrm{v}}}{{\gamma }_{\mathrm{v}}+1}=\frac{1}{v^2}\left(1-\frac{1}{{\gamma }_{\mathrm{v}}}\right)$$which I proved in the complete answer above and here.