A collision is a short interaction in which objects exert large forces on one another over a limited time. If the net external impulse on the chosen system is negligible during that interval, the system's total momentum is conserved:

$$\mathbf P_i=\mathbf P_f.$$

Momentum conservation alone does not determine what kind of collision occurred. Collision types are distinguished by what happens to the system's total kinetic energy.

Elastic collisions

A collision is elastic when total kinetic energy is the same before and after the collision:

$$K_i=K_f.$$

Both momentum and kinetic energy can then be used as constraints on the final motion.

Inelastic collisions

A collision is inelastic when kinetic energy is not conserved. In ordinary impacts involving deformation or friction,

$$K_f<K_i,$$

with some initial kinetic energy becoming internal energy, deformation, heat, or sound. Total energy is still conserved; only kinetic energy has changed form.

An interaction can also release stored internal energy so that

$$K_f>K_i.$$

Such events are often called superelastic collisions. They do not violate energy conservation: the additional kinetic energy comes from internal, chemical, elastic, or other stored energy.

Perfectly inelastic collisions

A perfectly inelastic collision is the limiting case in which the colliding objects stick together and move with one common final velocity. In one dimension,

$$m_1u_1+m_2u_2=(m_1+m_2)v_f,$$

so

$$v_f=\frac{m_1u_1+m_2u_2}{m_1+m_2}.$$

Here $u_1$ and $u_2$ are the initial velocities.

Example

A $2,\mathrm{kg}$ cart moving at $4,\mathrm{m/s}$ collides with a stationary $1,\mathrm{kg}$ cart and they stick together. Momentum conservation gives

$$v_f=\frac{(2)(4)+(1)(0)}{2+1}=\frac83\approx2.67,\mathrm{m/s}.$$

The initial kinetic energy is

$$K_i=\frac12(2)(4^2)=16,\mathrm J,$$

while the final kinetic energy is

$$K_f=\frac12(3)\left(\frac83\right)^2\approx10.7,\mathrm J.$$

About $5.3,\mathrm J$ of kinetic energy has been converted into other forms. The momentum remains conserved even though the kinetic energy does not.

The central distinction is therefore:

• momentum conservation follows from negligible external impulse on the system;
• elasticity describes whether kinetic energy is also conserved during the collision.