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 m$_1$ and m$_2$. The particle m$_1$ is moving with speed v$_{1i}$, the subscript ‘i’ implying initial. We can cosider m$_2$ to be at rest. No loss of generality is involved in making such a selection. In this situation the mass m$_1$ collides with the stationary mass m$_2$ and this is depicted in Fig. 1.
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| Fig.1: 0 Collision of mass $m_1$, with a stationary mass $m_2$. |
The masses m$_1$ and m$_2$ fly-off in different directions. We shall see that there are relationships, which connect the masses, the velocities and the angles.
Elastic and Inelastic Collisions
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 :
∆$\mathbf{p_1} = \mathbf{F_{12}}$∆t
∆$\mathbf{p_2} = \mathbf{F_{21}}$∆t
where $\mathbf{F_{12}}$ is the force exerted on the first particle by the second particle. $\mathbf{F_{21}}$ is likewise the force exerted on the second particle by the first particle. Now from Newton’s third law, $\mathbf{F_{12}}=-\mathbf{F_{21}}$. This implies
$\Delta \mathbf{p_{1}}+\Delta \mathbf{p_2}=0$
The above conclusion is true even though the forces vary in a complex fashion during the collision time ∆t. Since the third law is true at every instant, the total impulse on the first object is equal and opposite to that on the second.
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. The intermediate case where the deformation is partly relieved and some of the initial kinetic energy is lost is more common and is appropriately called an inelastic collision.
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