Question

Two particles $A$ and $B$ are moving with uniform velocity as shown in Fig. given at $t=0$.
a. Will the two particles collide?
b. Find out the shortest distance between two particles.

Answer 1

Observing $A$ from the frame of $B$
Let us draw $\overrightarrow{v_{AB}}$,
${\overrightarrow{v}}_{AB}={\overrightarrow{v}}_A-{\overrightarrow{v}}_B={\overrightarrow{v}}_A+(-{\overrightarrow{v}}_B)$
$tan\theta =\frac{10}{20}=\frac{1}{2}$
Again, $tan\theta =\frac{AD}{CD}=\frac{AD}{40}=\frac{1}{2}$
$\Rightarrow AD=20\Rightarrow DO=10$
$\Rightarrow BC=10$
Hence, the shortest distance between the particles
$d_{short}=BC\cos \theta =10\cos \theta =\frac{10\times 2}{\sqrt{5}}=4\sqrt{5}m$
Since the closest distance is non-zero, therefore, the particles will not collide.