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 $\text{3}/\text{4th}$ 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)

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, $K_F=\frac{3}{4}{K}_1$ $\Rightarrow$ $\frac{1}{2}m{v}_1^{{}^{\prime }2}=\frac{1}{2}m{v}_2^{{}^{\prime }2}=\frac{3}{4}\left(\frac{1}{2}m{v}^2\right)$ Substituting the values, we get ${\left(\frac{1+e}{2}\right)}^2+{\left(\frac{1-e}{2}\right)}^2=\frac{3}{4}$ $\Rightarrow$ ${(1+e)}^2+{(1-e)}^2=3$ $\Rightarrow$ $2+2e^2=3\Rightarrow e^2=\frac{1}{2}\Rightarrow e=\frac{1}{\sqrt{2}}$