Question

Two spheres $A$ and $B$ of masses $m_1$ and $m_2$ respectively collide. $A$ is at rest initially and $B$ is moving with velocity $v/2$ along x-axis. After collision $B$ has a velocity in a direction perpendicular to the original direction. The mass $A$ moves after collision in the direction

• A. same as that of B
• B. opposite to that of B
• C. $\theta=tan^{-1}\bigg(\frac{1}{2}\bigg)$ to the x-axis
• D. $\theta=tan^{-1}\bigg(-\frac{1}{2}\bigg)$ to the x-axis

Answer 1

The Correct Option is D

There is no external force acting on the spheres. So linear momentum will be conserved.
Before the collision, In direction $x$, Linear momentum $= m _{2} v$....1
In direction $y$, Linear momentum $=0$
After the collision, spheres moves as shown in figure. Let velocity of sphere A is
$v _{1}$
In direction $x$, Linear momentum $= m _{1} v _{1} \cos (\theta) \ldots 2$
In direction y, Linear momentum $=\frac{ m _{2} v }{2}- m _{1} v _{1} \sin (\theta) \ldots \ldots .3$
Linear momentum will be conserved,
From equation 1 and $2, \Rightarrow m_{1} v_{1} \cos (\theta)=m_{2} v \ldots 4$
From the equation 2, $\Rightarrow \frac{ m _{2} v }{2}- m _{1} v _{1} \sin (\theta)=0 \Rightarrow m _{1} v _{1} \sin (\theta)=\frac{ m _{2} v }{2} \ldots \ldots 5$
Dividing equation 5 by $4, \Rightarrow \tan (\theta)=\frac{1}{2} \Rightarrow \theta=\tan ^{-1}\left(\frac{1}{2}\right)$