Question

A ball of mass m moving with a velocity u collides head on with a second ball of mass m at rest. Coefficient of restitution is e.
The ratio of the final and initial velocities of the first ball is

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

Answer 1

The correct option is D $\frac{1-e}{2}$

The linear momentum before the $collision$ is $P_i=mu+0=mu$

suppose after collision the first mass move with velocity $V_1$ and the second with $V_2$

then the net momentum after collision will be $P_f=m{V}_1+m{V}_2=m(V_1+V_2)$

momentum will be conserved as there is $no$ external force on the system of these balls.

so $P_i=P_f$ so $u=V_1+V_2$...........(1)

given that coefficient of restitution is $e$ so $e=\frac{V_2-V_1}{u-0}$ or $V_1-V_2=-eu$ ........(2)

adding the equation-1 and equation-2

we get $2V_1=(1-e)u$

so the ratio will be $\frac{V_1}{u}=\frac{1-e}{2}$