Question

Prove that the magnitude of the ratio of the difference in speeds after collision, to the difference in speeds before collision, for a one-dimensional elastic collision,equal to one.​

Answer 1

Answer:

Solution :

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 m1,m2 are moving with velocities u1,u2 along the straight line in same direction collided elastically .

Let their velocities after collision be v1andv2 .

According to the law of conservation of linear momentum

m1u1+m2u2=m1v1+m2v2

m1(u1−v1)=m2(v2−u2) ....(1)

According to law of conservation of kinetic energy

12m1u21+12m2u22=12m1v21+12m2v22

m1(u21−v21)=m2(v22−u22) ...(2)

Dividing eqn . (2) by (1)

u21−v21u1−v1=v22−u22v2−u2 (or)

u1+v1=v2+u2⇒u1−u2=v2−v1....(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'.