Two particles moving in a laboratory frame of reference along the same straight line with the same velocity `v=(3//4)c` strike against a stationary target with the time interval `Deltat=50ns`. Find the proper distance between the particles prior to their hitting the target.

Explanation

In the frame K in which the particles are at rest, their positions are A and B whose coordinates may be taken as,
`A: (0, 0, 0), B=(l_0, 0, 0)`
In the frame `K^'` with respect to which K is moving with a velocity v the coordinates of A and B at time `t^'` in the moving frame are
`A=(vt^', 0, 0) B=(l_0sqrt(1-beta^2)+vt^',0,0), beta=v/c`
Suppose B hits a stationary target in `K^'` after time `t_B^'` while A hits it after time `t_B+Deltat`. Then,
`l_0sqrt(1-beta^2)+vt_B^'=v(t_B^'+Delta t)`
So, `l_0(vDelta t)/(sqrt(1-v^2//c^2))`