How Do Inelastic Collisions Work?

[Greg Bernhardt]

Definition/Summary

A collision is said to be inelastic if the total kinetic energy of all the bodies involved in the collision changes.

So conservation of kinetic energy does not apply.

But conservation of momentum, and of angular momentum, does apply to all unrestrained collisions.

For restrained collisions, conservation of momentum applies in any direction along which there is no impulsive restraining force, and conservation of angular momentum applies about any axis about which there is no impulsive restraining torque.

Most collisions are inelastic.

Equations

$$-e = \frac{v_2 - v_1}{u_2 - u_1}$$

e is the coefficient of restitution.

$v_1$ is the scalar final velocity of the first object after impact.

$v_2$ is the scalar final velocity of the second object after impact.

$u_1$ is the scalar initial velocity of the first object before impact.

$u_2$ is the scalar initial velocity of the second object before impact.

Conservation of momentum:
$$m_1 u_1 +m_2 u_2 = m_1 v_1 +m_2 v_2$$

If an object is bounced of a stationary object:

$$e = \sqrt{ \frac{h}{H} }$$
Where h is the height of the bounce
And H is the height the object was dropped from.

Extended explanation

In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy, such as heat sound and vibration.

Of the two principles which suffice to describe an elastic collision (conservation of kinetic energy and conservation of momentum), only conservation of momentum is available for an inelastic collision.

Therefore some other information is needed, and this is often available in the form of a Coefficient of Restitution.

Explosions:

An explosion can be treated as an inelastic collision.

Centre of mass:

In all collisions, momentum is conserved, and therefore the velocity of the centre of mass of the bodies in a collision is always the same just after the collision as it was just before.

In other words: the centre of mass instantaneously obeys Newton's first law.

For example, the trajectory of a rocket in which there is a tremendous explosion which is contained inside the rocket will not be altered by the explosion.​

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[AaronS]

Thanks for the overview of inelastic collisions