Question

A particle of mass m travelling with velocity $v$ and kinetic energy $E$ collides elastically to another particle of mass $nm$, at rest. What is the fraction of the total energy retained by the particle of mass $m$?

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

Answer 1

Answer: (C)

${\left(\frac{n-1}{n+1}\right)}^2$

Given : $m_1=m$ $m_2=nm$

Let the velocities of $A$ and $B$ after the collision be $v_1$ and $v_2$ respectively.

initial velocity of A before collision is $v$ and that of B is zero.

Initial kinetic energy of the system $E=\frac{1}{2}m{v}^2$

Using $v_1=\frac{(m_1-e{m}_2)u_1+(1+e)m_2{u}_2}{m_1+m_2}$

$v_1=\frac{(m-1\times nm)v+(1+1)nm\left(0\right)}{m+nm}$ $⟹v_1=\frac{(1-n)}{(1+n)}v$

Thus Kinetic energy of A after the collision $E_A^{\prime }=\frac{1}{2}\left(m\right)v_1^2$

$E_A^{\prime }=\frac{1}{2}\left(m\right)\frac{(1-n{)}^2{v}^2}{(1+n{)}^2}$ $⟹E_A^{\prime }=\frac{1}{2}m{v}^2(\frac{1-n}{1+n}{)}^2$

Fraction of total kinetic energy retained by A $\frac{E_A^{\prime }}{E}=(\frac{n-1}{n+1}{)}^2$