Question

A sphere of mass $m$ moving with a constant velocity $u$ hits another stationary sphere of the same mass and of coefficient of restitution $\left(e\right)$. The ratio of velocities of the two spheres, after collision will be

• A. $\frac{1-e}{1+e}$
• B. $\frac{e}{e+1}$
• C. $2/e$
• D. $\frac{e+1}{2e}$

Answer 1

The correct option is A $\frac{1-e}{1+e}$

$\text{Step 1: Apply momentum conservation}$

Since, there is no external force acting on the system, therefore momentum is conserved.

Let $v_1$ and $v_2$ be the velocities after collision

$P_i=P_f$

$\Rightarrow mu+0=m{v}_1+m{v}_2$

$\Rightarrow u=v_1+v_2$ $....\left(1\right)$

$\text{Step 2: Coefficient of restitution (e)}$

Coefficient of restitution is defined as:

$e=\frac{\text{Relative velocity of separation(after collision)}}{\text{Relative velocity of approach(before collision)}}=\frac{v_2-v_1}{u_1-u_2}$

$\Rightarrow e=\frac{v_2-v_1}{u-0}\Rightarrow v_2-v_1=eu$ $....\left(2\right)$

$\text{Step 3: Solving above equations}$

Adding eq $\left(1\right)$ and $\left(2\right)\Rightarrow$ $2v_2=(1+e)u$ $\Rightarrow v_2=\frac{(1+e)}{2}u$ $....\left(3\right)$

Subtracting eq $\left(2\right)$ from $\left(1\right)\Rightarrow$ $2v_1=(1-e)u$ $\Rightarrow v_1=\frac{(1-e)}{2}u$ $....\left(4\right)$

Dividing eq $\left(4\right)$ by $\left(3\right)$$\Rightarrow \frac{v_1}{v_2}=\frac{1-e}{1+e}$

Hence, Option(A) is correct.