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Homework Help: Reflection of particle between Inertial frames in SR

  1. Jul 14, 2015 #1
    1. The problem statement, all variables and given/known data
    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'r. 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.

    2. Relevant equations
    Velocity addition, lorentz contraction

    3. 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

    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=-uyi'-uxj'.
    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

    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.
  2. jcsd
  3. Jul 15, 2015 #2


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  4. Jul 15, 2015 #3
    By the principle of relativity θi = θr must be true, but θ may change, now the only difference between the two from is the tilt of the wall (let's forget about time for a while), I took at what you did and I totally agree with it, the great conclusion here is that angle is a frame dependant due to the contraction which itself is due to the relativity of sumiltaneity :)
  5. Jul 15, 2015 #4
    thank you :)
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