Ans:

Let us consider two colliding bodies of masses \( m_1 \) and \( m_2 \) moving with velocity \( u_1 \) and \( u_2 \) before collision. Let \( v_1 \) and \( v_2 \) be their respective velocities after perfectly elastic collision.

According to the principle of conservation of linear momentum we can write,

\( m_1u_1+m_2u_2=m_1v_1+m_2v_2\tag{1} \)

and according to the conservation of kinetic energy we can write,

\( \frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2=\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2 \)

or, \( m_1(u_1^2-v_1^2)=m_2(v_2^2-u_2^2)\tag{2} \)

Deviding equation (2) by equation (1) we get

\( u_1+v_1=u_2+v_2 \)

or, \( v_2=u_1+v_1-u_2\tag{3} \)

From equations (1) and (3) we get,

\( m_1(u_1-v_1)=m_2(u_1+v_1-2u_2) \)

or, \( (m_1+m_2)v_1=(m_1-m_2)u_1+2m_2u_2 \)

or, \( \displaystyle{v_1=\frac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2}}\tag{4} \)

From equations (2) and (4) we get,

\( \displaystyle{v_2=u_1-u_2+\frac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2}} \)

= \( \displaystyle{\left(1+\frac{m_1-m_2}{m_1+m_2}\right)u_1+\left(\frac{2m_2}{m_1+m_2}\right)u_2} \)

= \( \displaystyle{\frac{2m_1u_1}{m_1+m_2}+\frac{-m_1+m_2}{m_1+m_2}u_2} \)

or, \( \displaystyle{v_2=\frac{2m_1u_1+(m_2-m_1)u_2}{m_1+m_2}}\tag{5} \)

Therefore the total kinetic energy after collision is given by,

\( \frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2 \)

= \( \displaystyle{\frac{1}{2}m_1{\left(\frac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2}\right)}^2+\frac{1}{2}m_2{\left(\frac{2m_1u_1+(m_2-m_1)u_2}{m_1+m_2}\right)}^2} \)

= \( \displaystyle{\frac{m_1}{2{(m_1+m_2)}^2}\left[{(m_1-m_2)}^2u_1^2+4m_2^2u_2^2+4m_2u_1u_2(m_1-m_2)\right]\\+\frac{m_2}{2{(m_1+m_2)}^2}\left[{(m_2-m_1)}^2u_2^2+4m_1^2u_1^2+4m_1u_1u_2(m_2-m_1)\right]} \)

= \( \displaystyle{\frac{1}{2{(m_1+m_2)}^2}\left[m_1{(m_1-m_2)}^2u_1^2+4m_1m_2^2u_2^2+m_2{(m_2-m_1)}^2u_2^2+4m_2m_1^2u_1^2\right]} \)

= \( \displaystyle{\frac{1}{2{(m_1+m_2)}^2}\left[m_1u_1^2(m_1^2m_2^2+2m_1m_2)+m_2u_2^2(m_1^2+m_2^2+2m_1m_2)\right]} \)

= \( \displaystyle{\frac{1}{2{(m_1+m_2)}^2}\left[m_1u_1^2{(m_1+m_2)}^2+m_2u_2^2{(m_1+m_2)}^2\right]} \)

= \( \displaystyle{\frac{{(m_1+m_2)}^2}{2{(m_1+m_2)}^2}(m_1u_1^2+m_2u_2^2)} \)

= \( \displaystyle{\frac{1}{2}m_1u_1^2+\frac{1}{2}m_2u_2^2} \)

= total kinetic energy before collision