Ball \(A\) moving with momentum \(p\) undergoes a one dimensinal collision with a stationary ball \(B\) of…
Ball \(A\) moving with momentum \(p\) undergoes a one dimensinal collision with a stationary ball \(B\) of the same mass. During the collision ball \(B\) imparts an impulse \(I\) to ball \(A\). The coefficient of restitution is
\(\frac{2 I}{p}\)
\(\frac{2 I}{p}-1\)
\(\frac{I}{p}+1\)
\(\frac{2 I}{p}+1\)
Solution
Let $m$ be the mass of each ball. Let $u_{1}$ and $u_{2}$ be the velocities of $A$ and $B$ before collision and $v_{1}$ and $v_{2}$ are after collision. Then $u_{1}=\frac{p}{m}$ and $u_{2}=0$ (given).
Impulse = change in momentum. It is given that $I=$ change in momentum of $B$.
$\begin{aligned}
&=m v_{2}-0 \\
\Rightarrow \quad & v_{2} =\frac{I}{m}
\end{aligned}$
Therefore, $v_{1}=\frac{p-I}{m}$.
Now
$\begin{aligned}
e &=\frac{v_{2}-v_{1}}{u_{1}-u_{2}} \\
&=\frac{\left(\frac{I}{m}-\frac{p-I}{m}\right)}{\left(\frac{p}{m}-0\right)}=\frac{2 I}{p}-1
\end{aligned}$
So the correct choice is (b).