- #1

cpsinkule

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## Homework Statement

Suppose a frame S' is related to S by a boost in the y direction by

*v.*Imagine a wall is at rest in the S' frame along the line

*y'=-x'.*Consider a particle moving in the x'y' plane that strikes the surface and is reflected by the usual law of reflection θ'

_{i}

*=θ'*. Find the relation between the incident and reflected angles in the S frame as well as the initial and final velocity of the particle in the S frame.

_{r}## Homework Equations

Velocity addition, lorentz contraction

## The Attempt at a Solution

I assumed for simplicity that, although the wall is in S' and moving, I would always be able to redifine the S origin so that the wall passes through the origin in both frames. Now the wall itself will be contracted in the y direction, so the equation of the wall in the S frame is

*y/β=-x*(I'm using beta for the lorentz factor because gamma looks too similar to y). To find the angle of reflection and incidence in the S' frame I need a normal vector to the wall, which is easily obtained to be 1/√2*(

**i'**+

**j'**). Now the velocity of the particle in the S' frame is

**=u**

u'

u'

_{x}

**i'**+u

_{y}

**j'**

After striking the wall and being reflected, the particle's velocity will be reflected and reversed about the normal to the wall so that after contact the new velocity will be

**u**=-u

_{y}

**i'**-u

_{x}

**j'.**

Now the incident and reflected angles can be calculated (and are equal in this frame) by cosθ=

**n*u/|u|**

where

**n**is the unit normal to the wall 1/√2*(

**i'**+

**j'**)

To calculate the angles in the S frame we have to transform the initial and final velocities of the particles and then dot them with the NEW normal to the wall which can be obtained from

*y/β=-x,*remembering that I'm using beta for the lorentz factor instead of gamma. The normal to the wall in this frame can be considered by a point on the perpendicular line y=x/β considering when x=1 y=1/β the normalized unit vector along this line is then

(

**i**+1/β

**j**)/((1/β)

^{2}+1)

^{1/2}

Once I transform the above velocities via the addition formula and dot it with this normal vector, will I arrive at the correct answer? Is there an easier way to arrive at this result if it is correct? Forgive me for not typing the entire solution, but I feel like this is already in the realm of tldr.