Question

Two relativistic particles move at right angles to each other in a laboratory frame of reference, one with the velocity `v_1` and the other with the velocity `v_2`. Find their relative velocity.

Answer 1

The approach velocity is defined by
`vecV_(approach)=(dvecr_1)/(dt)-(dvecr_2)/(dt)=V_1-vecV_2`
in the laboratory frame. So `V_(approach)=sqrt(v_1^2+v_2^2)`
On the other hand, the relative velocity can be obtained by using the velocity addition formula and has the componets
`[-v_1, v_2sqrt(1-(v_1^2/c^2))]` so `V_r=sqrt(v_1^2+v_2^2-(v_1v_2^2)/(c^2))`