Discuss elastic collision in one dimension. Derive an
expression for velocities of two bodies after such a
collision.
Consider two bodies of masses m1 and m2 moving with velocities u1 and u2 moving
in the same straight line colliding with each other. Let their velocities be v1 and v2
after the collision.
Since momentum remains conserved in an elastic collision, therefore
m1u1 m2u2 m1v1 m2 v 2
m1u1 m1v1 m2 v 2 m2u2
m1 u1 v1 m2 v 2 u2 .......(i)
As kinetic energy is also conserved in elastic collision therefore
1 1 1 1
m1u21 m2u22 m1v12 m2 v 22
2 2 2 2
1 1 1 1
m1u12 m1v12 m2 v 22 m2u22
2 2 2 2
1 1
m1 u12 v12 m2 v 22 u22
2 2
1 1
m1 u1 v1 u1 v1 m2 v 2 u2 v 2 u2 .......(ii)
2 2
From (i) and (ii), we get
m1 u1 v1 u1 v1 m 2 v 2 u2 v 2 u 2
m1 u1 v1 m2 v 2 u 2
u1 v1 v 2 u2
u1 u2 v 2 v1 .............(iii)
Thus, relative velocity of approach = relative velocity of separation
Since
expression for velocities of two bodies after such a
collision.
Consider two bodies of masses m1 and m2 moving with velocities u1 and u2 moving
in the same straight line colliding with each other. Let their velocities be v1 and v2
after the collision.
Since momentum remains conserved in an elastic collision, therefore
m1u1 m2u2 m1v1 m2 v 2
m1u1 m1v1 m2 v 2 m2u2
m1 u1 v1 m2 v 2 u2 .......(i)
As kinetic energy is also conserved in elastic collision therefore
1 1 1 1
m1u21 m2u22 m1v12 m2 v 22
2 2 2 2
1 1 1 1
m1u12 m1v12 m2 v 22 m2u22
2 2 2 2
1 1
m1 u12 v12 m2 v 22 u22
2 2
1 1
m1 u1 v1 u1 v1 m2 v 2 u2 v 2 u2 .......(ii)
2 2
From (i) and (ii), we get
m1 u1 v1 u1 v1 m 2 v 2 u2 v 2 u 2
m1 u1 v1 m2 v 2 u 2
u1 v1 v 2 u2
u1 u2 v 2 v1 .............(iii)
Thus, relative velocity of approach = relative velocity of separation
Since
