In physics we study motion (change in position). At the same time, we try to discover physical quantities, which do not change in a physical process. The laws of momentum and energy conservation are typical examples. In this section we shall apply these laws to a commonly encountered phenomena, namely collisions. Several games such as billiards, marbles or carrom involve collisions. We shall study the collision of two masses in an idealised form. Consider two masses m1 and m2. The particle m1 is moving with speed v1i, the subscript ‘i’ implying initial. We can consider m2 to be at rest. No loss of generality is involved in making such a selection. In this situation the mass m1 collides with the stationary mass m2 and this is depicted in Fig. 5.10.

In all collisions the total linear momentum is conserved; the initial momentum of the system is equal to the final momentum of the system. One can argue this as follows. When two objects collide, the mutual impulsive forces acting over the collision time ∆t cause a change in their respective momenta: ∆p1 = F12 ∆t ∆p2 = F21 ∆t where F12 is the force exerted on the first particle by the second particle. F21 is likewise the force exerted on the second particle by the first particle. Now from Newton’s third law, F12 = − F21. This implies ∆p1 + ∆p2 = 0.

On the other hand, the total kinetic energy of the system is not necessarily conserved. The impact and deformation during collision may generate heat and sound. Part of the initial kinetic energy is transformed into other forms of energy. A useful way to visualise the deformation during collision is in terms of a ‘compressed spring’. If the ‘spring’ connecting the two masses regains its original shape without loss in energy, then the initial kinetic energy is equal to the final kinetic energy but the kinetic energy during the collision time ∆t is not constant. Such a collision is called an elastic collision. On the other hand the deformation may not be relieved and the two bodies could move together after the collision. A collision in which the two particles move together after the collision is called a completely inelastic collision.

Consider next an elastic collision. Using the above nomenclature with θ1 = θ2 = 0, the momentum and kinetic energy conservation equations are m1v1i = m1v1f + m2v2f (5.23) 2 2 2 1 1 1 1 2 21 1 1 2 2 2 i f f m v m v m v= + (5.24).

If the initial velocities and final velocities of both the bodies are along the same straight line, then it is called a one-dimensional collision, or head-on collision. In the case of small spherical bodies, this is possible if the direction of travel of body 1 passes through the centre of body 2 which is at rest. In general, the collision is two-dimensional, where the initial velocities and the final velocities lie in a plane.