(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A particle leaves the spatial originPofOat time [itex]t=0[/itex] and constant velocity. After a timetas measured byO, a second particleBleavesPat a different constant velocity and in pursuit ofA.BcatchesAafter proper-timetas measured byB. Show that the rapidity ofBwith respect toOis twice the rapidity ofAwith respect toO.

2. Relevant equations

I assign the coordinates [itex](t', x')[/itex] within this frame (slightly counter-intuitive I know, but the problem has already denoted some variables as unprimed, which I might otherwise want to use). I'm not adept at drawing diagrams in LaTeX, so I won't bother. However, as I envisaged the problem, the worldlines of particles A and B are as follows:

[tex]t'_A(x')=x'/v_A[/tex]

[tex]t'_B(x')=t + x'/v_B[/tex]

I label the event where they meet as [itex](t'_1,x'_1)[/itex]. So we have the equality

[tex]x'_1 \left(\frac{1}{v_A}-\frac{1}{v_B}\right)=t\,\,\,\,\,\,\,(1)[/tex]

We have the fact that the proper-time as measured by the particleBalso happens to bet. I could think of two ways to write that:

[tex]t=\gamma_B (t'_1-t)\,\,\,\,\,\,\,(2)[/tex]

and also as (which I think is correct)

[tex]c^2(t'_1-t)^2-x'_1^2=c^2 t^2\,\,\,\,\,\,\,(3)[/tex]

I list some handy rapidity based relations (where the rapidity, [itex]\phi,[/itex] is related to the velocity, v,):

[tex]\beta = \frac{v}{c} = \tanh(\phi)[/tex]

[tex]\gamma = (1-\beta^2)^{-1/2} = \cosh(\phi)[/tex]

[tex]\beta \gamma = \sinh(\phi)[/tex]

3. The attempt at a solution

My method of attack was to use the equation (1) and either (2)/(3) and obtain some relation between the velocities of both particles, which I could then take the artanh of to get the rapidities.

Processing equation (1) gave me

[tex]t=x'_1\left(\frac{v_B - v_A}{v_A v_B}\right).[/tex]

Substituting this into (2) gave me

[tex]\left(\frac{v_B - v_A}{v_A v_B}\right) (1+\gamma_B)=\frac{t'_1}{x'_1}=\frac{1}{v_A}[/tex]

Playing around with fractions ended up giving me

[tex]v_A = v_B \left( \frac{\gamma_B}{1+\gamma_B} \right)[/tex]

Using the definitions/relations regarding rapidity above, I still couldn't get to the answer I wanted, which is

[tex]\phi_B = 2 \phi_A[/tex]

Does anyone have any ideas? Thanks for reading this long question!

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# Homework Help: SR: Rapidity of two particles in uniform motion

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