LINEAR MOMENTUM

4. COLLISION

1.4 Collisions

A collision occurs when two or more objects come into contact with each other and exchange energy or momentum. There are two principal types of collisions – the elastic and the inelastic collisions.

1.4.1 Elastic collision

In an elastic collision, the total kinetic energy of the colliding objects is conserved. This means that the objects bounce off each other without any loss of kinetic energy. Thus on such collision, both the momentum and the kinetic energy are conserved.

In a perfectly elastic collision, relative speed of approach = relative speed of separation.

Let us consider two bodies of masses m₁ and m₂ moving with initial velocities u₁ and u₂ before collision and with final velocities v₁ and v₂ after collision in the same direction. If the collision is perfectly elastic, we can write two equations from the principles of conservation of momentum and conservation of kinetic energy. Hence

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²

Two examples of collisions that are often very nearly perfectly elastic are the collisions of billiard balls and of molecules and atoms. In a perfectly elastic head-on collision between two bodies, the relative velocity of the two bodies is unchanged in magnitude but reversed in direction.

1.4.2 Inelastic collision

In an inelastic collision, the kinetic energy decreases after collision but the momentum is still conserved. The colliding bodies stick together and move as a unit after collision. This means that the velocities of the two bodies after collision are

                        v₁ =  v₂ = v

from conservation of linear momentum, we have

                        m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ = (m₁ + m₂)v

the kinetic energy of the system before collision is given by

                        KE₁ = ½m₁u₁² + ½m₂u₂²

And after collision, the kinetic energy is

                        KE₂ = ½m₁v₁² + ½m₂v₂² = ½(m₁ + m₂)v²

For a completely inelastic collision, the kinetic energy before collision is greater than the kinetic energy after collision.

1.4.3 Examples

1. A bullet of mass 120g is fired horizontally into a fixed wooden block with a speed of 20ms^-1. The bullet is brought to rest in the block in 0.1s by a constant resistance. Calculate the

(i) magnitude of the resistance.

(ii) distance moved by the bullet in the wood

2. A tractor of mass 5.0 x 10³kg is used to tow a car of mass 2.5 x 10³kg. The tractor moved with a speed of 3.0ms^-1 just before the towing rope becomes taut. Calculate the

(i) speed of the tractor immediately the rope becomes taut.

(ii) loss in KE of the system just after the car has started moving.

(iii) impulse in the rope when it jerks the car into motion.