Question

A ball of mass $m$ moving at a speed $v$ makes a head on collision with an identical ball at rest. The kinetic energy at the balls after the collision is $3/4$th of the original. What is the coefficient of restitution?

• A. $1/\sqrt{3}$
• B. $1/\sqrt{2}$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

Answer: (B)

$1/\sqrt{2}$

Since, we know that in the case, when $m_1=m_2$ and $v_1=0$
then, $v_1^{\prime }=\left(\frac{1+e}{2}\right)v_2$ and $v_2^{\prime }=\left(\frac{1-e}{2}\right)v_2$
Given that, $K_F=\frac{3}{4}{K}_i$
$\Rightarrow \frac{1}{2}m{v}_1^{{\prime }^2}=\frac{1}{2}m{v}_2^{{\prime }^2}=\frac{3}{4}\left(\frac{1}{2}m{v}^2\right)$
Substituting the values, we get
${\left(\frac{1+e}{2}\right)}^2+{\left(\frac{1-e}{2}\right)}^2=\frac{3}{4}$
$\Rightarrow {(1+e)}^2+{(1-e)}^2=3$
$\Rightarrow 2+2e^2=3\Rightarrow e^2=\frac{1}{2}\Rightarrow e=\frac{1}{\sqrt{2}}$