Question

Two particles start moving simultaneously with constant velocities $u_1$ and $u_2$ as shown in fig. First particle starts from $A$ along $AO$ and second starts from $O$ along $OM$. Find the shortest distance between them during their motion.

Answer 1

${\overrightarrow{u}}_1=-u_1\hat{j},{\overrightarrow{u}}_2\cos \theta j+u_2\sin \theta \hat{i}$
$\Rightarrow {\overrightarrow{u}}_2-{\overrightarrow{u}}_1{u}_2\sin \theta \hat{i}+(u_2\cos \theta +u_1)\hat{j}$
$\tan \alpha =\frac{u_2\sin \theta }{u_2\cos \theta +u_1}$
Shortest distance
$AB=x_0\sin \alpha =\frac{x_0{u}_2\sin \theta }{\sqrt{u_1^2+u_2^2+2u_1{u}_2\cos \theta }}$.