Question

In case of an elastic head on collision between two bodies, derive an expression for the final velocities of the bodies in terms of their masses and velocities before collision.

Answer 1

Head on elastic collision of two spheres:

i. Consider two rotating smooth bodies A and B of masses m₁ and m₂ respectively moving along the same straight line.

ii. Let \(\overrightarrow{u}_1\) = initial velocity of the sphere A before collision.

\(\overrightarrow{u}_2\) = initial velocity of the sphere B before collision.

\(\overrightarrow{\text{v}}_1\) = velocity of the sphere A after collision.

\(\overrightarrow{\text{v}}_2\) = velocity of the sphere B after collision.

iii. After the elastic collision, the spheres separate and move along the same straight line without rotation.

iv. According to the law of conservation of momentum,

According to the law of conservation of energy (as kinetic energy is conserved during elastic collision),

v. Since kinetic energy is a scalar quantity, the terms involved in the above equations are scalars.

vi. The equation (1) can be written in scalar form as,

vii. Also the equation (2) can be written as,

viii. Now dividing equation (4) by (3) we get,

ix. Comparing equation (3) and (5),

Substituting the value of v₁ in equation (5), we have

Equations, (6) and (7), represent the final velocities of two spheres after collision.