Question

A ball moving with a certain velocity collides. with another ball B of the same mass at rest. if the coefficient of restitution is e. The ratio of the velocities of A and B at collision is

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

Answer 1

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

Let, $u_A=$velocity of ball A before collision

$u_B=0m/s$ ball B is at rest before collision

$m_A=m_B=m\left(say\right)$

$e=$ coefficient of restitution

Let the ball A will move with velocity of $v_A$ and the ball B with $v_B$ after collision.

By conservation of linear momentum,

$m_A{u}_A+m_B{u}_B=m_A{v}_A+m_B{v}_B$

$m{u}_A+m.0=m{v}_A+m{v}_B$

$v_A+v_B=u_A$. . . . . . . . . .(1)

Now, the coefficient of restitution,

$e=\frac{v_B-v_A}{u_A-0}$

$v_B-v_A=e{u}_A$. . . . .(2)

Solving equation (1) and equation (2), we get

$v_A=\frac{u_A(1-e)}{2}$

$v_B=\frac{u_A(1+e)}{2}$

Ratio, $\frac{v_A}{v_B}=\frac{\frac{u_A(1-e)}{2}}{\frac{u_A(1+e)}{2}}=\frac{1-e}{1+e}$

The correct option is D.