Question

A ball of mass $m$ moving with speed $u$ undergoes a head-on elastic collision with a ball of mass $nm$ initially at rest, the ratio of the kinetic energies of the first ball to the second ball after collision is

• A. $\frac{(1-n{)}^2}{2n}$
• B. $\frac{(1-n{)}^2}{4n}$
• C. $\frac{(1+n{)}^2}{2n}$
• D. $\frac{(1+n{)}^2}{4n}$

Answer 1

The correct option is B $\frac{(1-n{)}^2}{4n}$

By conservation of momentum

$mu=m{u}_1+nm{u}_2\Rightarrow u=u_1+n{u}_2\to \left(1\right)$

Co-efficient of restitution

$\Rightarrow u-0=u_2-u_1\to \left(2\right)\left(2\right)in\left(1\right)\Rightarrow u_2=\frac{2u}{n+1}\&u_1=\frac{(1-n)u}{n+1}\therefore \frac{K{E}_{F1}}{K{E}_{F2}}=\frac{1/2m{u}_1^2}{1/2nm{u}_2^2}=\frac{u_1^2}{n{u}_2^2}=\frac{(1-n{)}^2}{4n}$