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Types of Collisions

For mathematical simplicity, two extreme cases of collision are considered. They are elastic and inelastic collisions and it can be described as follows.

Elastic Collisions

If there is no loss of kinetic energy during a collision it is called an elastic collision. The collision between subatomic particles is generally elastic. The collision between two steel or glass balls is nearly elastic.

Inelastic Collisions

If there is a loss of kinetic energy during a collision, it is called an inelastic collision. Since there is always some loss of kinetic energy in any collision, collisions are generally inelastic.

If the loss is negligibly small, the collision is very nearly elastic. Perfectly elastic collisions are not possible. If two bodies stick together, after colliding, the collision is perfectly inelastic, e.g. a bullet striking a block of wood and being embedded in it. The loss of kinetic energy usually results in heat or sound energy.

It may be remembered that the total momentum remains conserved in both elastic and inelastic collisions. Further, since the interacting forces become effectively zero after the collision, the potential energy remains the same both before and after the collision.

Calculation of Final Velocities of Colliding Bodies

The velocities of the particles after a collision can be calculated by solving their equations of motion if we know their velocities before the collision and the forces acting during the collision.

Unfortunately, we do not always know these interacting forces. Even if we do not know the forces involved, it is still possible to determine the final velocities in the following three cases:
  1. The collision is perfectly inelastic, i.e. the colliding body sticks to the target body after collision.
  2. The collision is elastic but one-dimensional, i.e. the bodies move in the same straight line before and after the collision.
  3. The collision is elastic and two-dimensional and the direction of motion of the bodies after collision is known.  We will consider these cases separately.

Inelastic Collision

Let m1 and m2 be the masses of the two colliding particles and u1 and u2 their respective velocities before collision. Then we have
Total momentum of particles before collision = m1u1+m2u2

After collision, the two particles stick together and move with a common velocity, say v.

Total momentum of particles after collision = Mass of the composite particle × common velocity = (m1+m2) v  

Since the momentum is always conserved in a collision, we have
Total momentum before collision = total momentum after collision

Thus knowing the masses of the two particles and their velocities before collision, we can calculate their common velocity after collision.

If the second particle is at rest initially (i.e.u2 = 0) then we have

In this case, we can easily see that there indeed is a loss in the kinetic energy. Now the kinetic energy of the particles before collision is given by (the second particle being at rest, u2 = 0)

But the kinetic energy of the particles after collision


< 1

This shows that kinetic energy after collision is less than that before collision. As mentioned earlier this loss in kinetic energy appears mostly in the form of heat and sound energy.

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