Question

A ball of mass $m$ travelling at a speed $v$ hits a rod of the same mass $m$ and length $L$ as shown in the figure below. After the collision , the ball sticks to the rod.Find the speed of the ball after collision.

• A. $\frac{4v}{5}$
• B. $\frac{3v}{5}$
• C. $\frac{2v}{5}$
• D. $\frac{v}{5}$

Answer 1

The correct option is A $\frac{4v}{5}$

Conserving the momentum we get,
$mv=m(v_{cm}+\omega \frac{L}{2})+m{v}_{cm}$
$2v=4v_{cm}+\omega L$ _____(i)

Conserving the angular momentum we get,
$mv\frac{L}{2}=m(v_{cm}+\omega \frac{L}{2})+\frac{m{l}^2\omega }{12}$

$\Rightarrow 3v=3v_{cm}+2\omega l$ ______(ii)

Solving (i) and (ii)
$v_{cm}=\frac{v}{5}$
and $\omega =\frac{6v}{5l}$
Speed of the ball wrt the ground
$v^{\prime }=v_{cm}+\omega \frac{l}{2}$
$v^{\prime }=\frac{4v}{5}$

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