Question

A collision occurs between two identical balls each of mass $m$ , moving with velocity $\overrightarrow{u_1}$ and $\overrightarrow{u_2}$ , colliding head-on. The coefficient of resolution is $0.5$ The energy lost in the collision is :

• A. $\frac{1}{4}m(\overrightarrow{u_1}+\overrightarrow{u_2}{)}^2$
• B. $\frac{1}{4}m(\overrightarrow{u_1}-\overrightarrow{u_2}{)}^2$
• C. $\frac{3}{16}m(\overrightarrow{u_1}-\overrightarrow{u_2}{)}^2$
• D. $\frac{1}{16}m(\overrightarrow{u_1}-\overrightarrow{u_2}{)}^2$

Answer 1

The correct option is C $\frac{3}{16}m(\overrightarrow{u_1}-\overrightarrow{u_2}{)}^2$

Resolution coefficient is the ratio of relative speed after the collision to the relative speed before the collision. Thus,

${\overrightarrow{v}}_f={\overrightarrow{u}}_1-{\overrightarrow{u}}_2$ and ${\overrightarrow{v}}_i=\frac{{\overrightarrow{u}}_1-{\overrightarrow{u}}_2}{2}$

Change in kinetic energy is,

$\Delta E=\frac{1}{2}m\left[{\overrightarrow{v}}_f^2-{\overrightarrow{v}}_i^2\right]$

$=\frac{1}{2}m\left[{\left({\overrightarrow{u}}_1-{\overrightarrow{u}}_2\right)}^2-{\left(\frac{{\overrightarrow{u}}_1-{\overrightarrow{u}}_2}{2}\right)}^2\right]$

$=\frac{3}{8}m{\left({\overrightarrow{u}}_1-{\overrightarrow{u}}_2\right)}^2$

This loss in kinetic energy is for both the balls. Therefore, kinetic energy loss of each ball is,

$\Delta E==\frac{3}{16}m{\left({\overrightarrow{u}}_1-{\overrightarrow{u}}_2\right)}^2$