Question

A bullet of mass $m$ is fired with the velocity of $50m{s}^{-1}$ at an angle $\theta$ with the horizontal. At the highest point of its trajectory, it collides head-on with a bob of mass $4m$ connected with a massless string of length $l=10/3m$ and gets embedded with the bob. After the collision, the string moves to an angle of ${120}^o$. What is the angle $\theta$

• A. ${\cos }^{-1}\left(\frac{1}{5}\right)$
• B. ${\cos }^{-1}\left(\frac{5}{4}\right)$
• C. ${\sin }^{-1}\left(\frac{4}{5}\right)$
• D. ${\sin }^{-1}\left(\frac{5}{4}\right)$

Answer 1

The correct option is A ${\cos }^{-1}\left(\frac{1}{5}\right)$

Given,
$m_{bullet}=m$
$m_{bob}=4m$
$\text{After Collision,}$
$\frac{1}{2}\left(5\right)m{v}^2=5mg\times (l+l\cos ({60}^{\odot })$
$v^2=g\times 3l$
$v=10m{s}^{{}_1}$
$\text{KE before collision = KE after collision}$
$\frac{1}{2}m{v}_i^2=\frac{1}{2}5m{v}^2$
$\frac{1}{2}m\left(50\cos \right(\theta ){)}^2=\frac{1}{2}5m\times 100$
$2500co{s}^2\theta =500$
$\theta ={\cos }^{-1}\left(\frac{1}{5}\right)$

Hence, Option $\text{A}$ is correct answer.