27. A moving body of mass m collides head on elastically, with a stationary body of mass3 m. Calculate the ratio of the momentum of the mass 3m, after the collision, to the momentum of mass m before the collision.
Answers
Answer:
A moving body of mass m collides head on elastically with a stationary body of mass 3 m.
To Find : ratio of the momentum of the mass 3m, after the collision, to the momentum of mass m before the collision.
Solution:
A moving body of mass m with velocity U₁
Momentum = mU₁
stationary body of mass 3 m at rest U₂ = 0
After collision
mass m with velocity V₁
mass 3m with velocity V₂
momentum of the mass 3m, after the collision = 3mV₂
Conservation of total momentum
mU₁ + 3mU₂ = m V₁ + 3mV₂
=> mU₁ = m V₁ + 3mV₂
=> U₁ = V₁ + 3V₂
As collision is elastic hence conservation of total kinetic energy
(1/2)mU₁² + (1/2)3mU₂² = (1/2)mV₁² + (1/2)3mV₂²
=> U₁² = V₁² + 3V₂²
=> (V₁ + 3V₂)² = V₁² + 3V₂²
=> V₁² + 9V₂² + 6V₁V₂ = V₁² + 3V₂²
=> 6V₂² + 6V₁V₂ = 0
=> V₂ + V₁ = 0
=> V₁ = - V₂
U₁ = V₁ + 3V₂
=> U₁ = - V₂ + 3V₂
=> U₁ = 2V₂
momentum of the mass 3m, after the collision = 3mV₂
momentum of mass m before the collision. mU₁ = 2mV₂
3mV₂ : 2mV₂
= 3 : 2
ratio of the momentum of the mass 3m, after the collision, to the momentum of mass m before the collision. = 3 : 2