LINEAR MOMENTUM
Linear Momentum as a course in physics aims to help students understand how objects behave when they are in motion and how they interact with each other, especially in terms of force, mass, and velocity.
4. COLLISION
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.