Question

The velocities of two bodies having masses $5kg$ and $10kg$ are $4m{s}^{-1}$ and $2m{s}^{-1}$ respectively along $x$ and $y$ axes. They collide and stick together. The compound body moves with velocity $v$ at angle $\theta$ with $x$ -axis. The value of $\theta$ is

• A. $30^{\circ }$
• B. $45^{\circ }$
• C. $60^{\circ }$
• D. $-75^{\circ }$

Answer 1

Answer: (B)

${\vec{p}}_{\text{initial }}=(5kg)\left(4\frac{m}{s}\right)\hat{i}+(10kg)\left(2\frac{m}{s}\right)\hat{j}$
$=(20\hat{i}+20\hat{j})\frac{kgm}{s}$
${\vec{p}}_{\text{final }}=(15kg)(v\cos \theta )\hat{i}$
$+(15kg)(v\sin \theta )\hat{j}$
$=(15v\cos \theta )\hat{i}+(15v\sin \theta )\hat{j}$
For collision, ${\bar{p}}_{\text{initial }}={\bar{p}}_{\text{final }}$
$(20\hat{i}+20\hat{j})=(15v\cos \theta )\hat{i}+(15v\sin \theta )\hat{j}$
$\Rightarrow 15v\sin \theta =20$
$\Rightarrow 15v\cos \theta =20$
Dividing the two equations, $\tan \theta =1$
$\Rightarrow \theta =45^{\circ }$.