# Elastic collision

 related topics {math, energy, light} {math, number, function}

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Elastic collisions occur only if there is no net conversion of kinetic energy into other forms. This definition applies to close encounters between a spacecraft and a gravitating body ( see gravity assist) as well as to actual collisions between individual atoms etc.

During the collision of small objects, kinetic energy is first converted to potential energy associated with a repulsive force between the particles (when the particles move against this force, i.e. the angle between the force and the relative velocity is obtuse), then this potential energy is converted back to kinetic energy (when the particles move with this force, i.e. the angle between the force and the relative velocity is acute).

The collisions of atoms are elastic collisions (Rutherford backscattering is one example).

The molecules—as distinct from atoms—of a gas or liquid rarely experience perfectly elastic collisions because kinetic energy is exchanged between the molecules’ translational motion and their internal degrees of freedom with each collision. At any one instant, half the collisions are, to a varying extent, inelastic collisions (the pair possesses less kinetic energy in their translational motions after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Averaged across the entire sample, molecular collisions can be regarded as essentially elastic as long as black-body photons are not permitted to carry away energy from the system.

In the case of macroscopic bodies, elastic collisions (except for near encounters with a gravitating body) are an ideal never fully realized, but approximated by the interactions of objects such as billiard balls.

When considering energies, possible rotational energy before and/or after a collision may also play a role.

## Contents

### One-dimensional Newtonian

Consider two particles, denoted by subscripts 1 and 2. Let mi be the masses, ui the velocities before collision and vi the velocities after collision.

The conservation of the total momentum demands that the total momentum before the collision is the same as the total momentum after the collision, and is expressed by the equation

Likewise, the conservation of the total kinetic energy is expressed by the equation

These equations may be solved directly to find vi when ui are known or vice versa. However, the algebra[1] can get messy. A cleaner solution is to first change the frame of reference such that one of the known velocities is zero. The unknown velocities in the new frame of reference can then be determined and followed by a conversion back to the original frame of reference to reach the same result. Once one of the unknown velocities is determined, the other can be found by symmetry.