The distance between two moving particles P and Q at any time is a. If$v_r$ be their relative velocity and if u and v be the components of $v_r$ , along and perpendicular to PQ. The closest distance between P and Q and time that elapses before they arrive at their nearest distance is

Assuming P to be at rest, particle Q is moving with velocity $v_r$ , in the direction shown in figure. components of $v_r$ along and perpendicular to PQ are u and v respectively, In the figure $\sin \alpha =\frac{u}{v_r},\cos \alpha =\frac{v}{v_r}$

The closest distance between the particles is PR.

$S_{\min }=PR=PQ\cos \alpha =\left(a\right)\left(\frac{v}{v_r}\right)\Rightarrow S_{\min }=\frac{av}{v_r}$

Time after which they arrive at their nearest distance is

$t=\frac{QR}{v_r}=\frac{\left(PQ\right)\sin \alpha }{v_r}=\frac{\left(a\right)\left(\frac{v}{v_r}\right)}{v_r}=\frac{au}{v_r^2}$