Question

The velocities of two bodies having masses $5 \,kg$ and $10 \,kg$ are $4 \,ms ^{-1}$ and $2 \, ms ^{-1}$ 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)

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