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}^{'}=\left( \frac{1+e}{2} \right){{v}_{2}} $ and $ v_{2}^{'}=\left( \frac{1-e}{2} \right){{v}_{2}} $ Given that, $ {{K}_{F}}=\frac{3}{4}{{K}_{1}} $ $ \Rightarrow $ $ \frac{1}{2}mv_{1}^{'2}=\frac{1}{2}mv_{2}^{'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+2{{e}^{2}}=3\Rightarrow {{e}^{2}}=\frac{1}{2}\Rightarrow e=\frac{1}{\sqrt{2}} $