Chapter 9
Linear Momentum and its Conservation
9.3 Elastic and Inelastic Collisions
In this section we consider the effect of a collision between two objects. There are two types of collision: inelastic and elastic collisions. Inelastic collision is a collision in which the colliding objects become distorted, generate heat, and possibly stick together. Hence, in case of inelastic collision the total linear momentum is conserved but both kinetic energy and mechanical energy are not conserved. . While in of elastic collision both linear moment and kinetic energy are both conserved.
In any system of colliding objects, usually we know the initial velocities of the two objects before the collision, and our goal is to apply conservation laws or laws of motion to find the velocities after collision:
1. The law of conservation of momentum must hold during any collision in which internal forces act, and it can be applied even if we do not know the force.
2. Total energy is also conserved: the initial total energy of the colliding particles is equal to the final total energy of the products.
That is, even without the detail knowledge of interaction, we can use the conservation of momentum and the conservation of energy in many cases to predict the results of the collision.
9.3.1 Inelastic Collisions
When two objects stick together after collision, the collision is said to be completely inelastic. Let us now consider two objects moving towards each as shown in Fig. 9.3(a). After sometimes they come into head-on collision, stick and move together as shown in Fig. 9.3(b).
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Fig. 9.3 Collision of two particles.
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Applying the law of conservation of momentum from Eq. (9.12) to the system before and after collision i.e.,
(momentum conservation) (9.19)
We can calculate the change in kinetic energy (KE) as follows:
(change in kinetic energy)
(9.20)
The use of Eq. (9.20) in solving problems should
result in a non-zero result, 
.
That is in of inelastic collision the total linear momentum is conserved
but not kinetic energy.
9.3.2 Elastic Collision
Elastic collision is a collision in which colliding objects rebound without lasting deformation or the generation of heat. In such case the objects are side to undergo a perfectly elastic collision. That is, in case of elastic collision both momentum and KE are conserved. Now let us consider again two particles moving towards each, they then collide and move in away from each other as shown in Fig. 9.4.
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Fig. 9.4 Elastic collision of two particles.
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From the conservation of linear momentum, Eq. (9.12):
(9.21)
Recall the conservation of KE:
(kinetic energy conservation)
(9.22)
.
That is, in case of elastic
collision both momentum and KE are conserved
