Give the definition of coefficient of restitution .Prove that for perfect elastic collision of two bodies moving in the same line , the kinetic energy is conserved.

Explanation

1 st part:
2nd part: Suppose `m_1 and m_2` are the masses of two particles which are moving with velocities `u_1 and u_2` respectively `(u_1gt u_2` in the same direction along a straight line. They collide elastically , and after collision, move along the same direction with velocities `v_1 and v_2` respectively.
So, the coefficient of restitution,`e=(v_2-v_1)/(u_1-u_2)`
For elastic collision ,e=1
`or, v_2-v_1=u_1-u_2`
`or, u_1+v_1=u_2+v_2`...... (1)
As no external force is applied on the system, from the law of conservation of momentum,
total momentum before collision= total momentum after collision
`or, m_1u_1+m_2u_2=m_1v_1+m_2v_2`
`or, m_1(v_1-u_1)=m_2(u_2-v_2)........(2)`
Multiplying equations (1) and (2) ,
`m_1(v_1^2-u_1^2)=m_2(u_2^2-v_2^2)`
`or, 1/2 m_1(v_1^2-u_1^2) =1/2 m_2(u_2^2-v_2^2)`
`or, 1/2 m_1v_1^2+1/2 m_2v_2^2=1/2 m_1u_1^2+1/2m_2u_2^2`
`therefore` Kinetic energy after collision = kinetic energy before collision.