Show that in the case of one dimensional elastic collision , the relative velocity of approach of two colloding bodies before collision is equal to the relative velocity of separation after collosion .

Explanation

To show relative velocity of approach of two colliding bodies before collision is equal to relative velocity of separation after collision .
Let two bodies of masses `m_(1),m_(2)` are moving with velocities `u_(1),u_(2)` along the straight line in same direction collided elastically .
Let their velocities after collision be `v_(1)andv_(2)` .

According to the law of conservation of linear momentum
`m_(1)u_(1)+m_(2)u_(2)=m_(1)v_(1)+m_(2)v_(2)`
`m_(1)(u_(1)-v_(1))=m_(2)(v_(2)-u_(2))" " ....(1)`
According to law of conservation of kinetic energy
`(1)/(2)m_(1)u_(1)^(2)+(1)/(2)m_(2)u_(2)^(2)=(1)/(2)m_(1)v_(1)^(2)+(1)/(2)m_(2)v_(2)^(2)`
`m_(1)(u_(1)^(2)-v_(1)^(2))=m_(2)(v_(2)^(2)-u_(2)^(2))" " ...(2)`
Dividing eqn . (2) by (1)
`(u_(1)^(2)-v_(1)^(2))/(u_(1)-v_(1))=(v_(2)^(2)-u_(2)^(2))/(v_(2)-u_(2))` (or)
`u_(1)+v_(1)=v_(2)+u_(2)rArru_(1)-u_(2)=v_(2)-v_(1)....(3)`
i.e, relative velocity of approach of the two bodies before collision = relative velocity of separation of the two bodies after collision . So coefficient of restitution is equal to 'l'.