Question

Two particles moving in a laboratory frame of reference along the same straight line with the same velocity $v=\left(\frac{3}{4}\right)c$ strike against a stationary target with the time interval $\Delta t=50ns$. Find the proper distance between the particles prior to their hitting the target.

Answer 1

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 $K^{\prime }$ with respect to which $K$ is moving with a
velocity $v$ the coordinates of $A$ and $B$ at time $t^{\prime }$
in the moving frame are
$A=(v{t}^{\prime },0,0)B=(l_0\sqrt{1-{\beta }^2}+v{t}^{\prime },0,0),\beta =\frac{v}{c}$
Suppose
$B$ hits a stationary target in $K^{\prime }$ after time
${t^{\prime }}_B$ while $A$ hits it after time $t_B+\Delta t$. Then,
$l_0\sqrt{1-{\beta }^2}+v{t^{\prime }}_B=v({t^{\prime }}_B+\Delta t)$
So, $l_0=\frac{v\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$