An *elastic collision* is a situation where multiple objects collide and the total kinetic energy of the system is conserved, in contrast to an *inelastic collision*, where kinetic energy is lost during the collision. All types of collision obey the law of conservation of momentum.

In the real world, most collisions result in loss of kinetic energy in the form of heat and sound, so it's rare to get physical collisions that are truly elastic. Some physical systems, however, lose relatively little kinetic energy so can be approximated as if they were elastic collisions. One of the most common examples of this is billiard balls colliding or the balls on Newton's cradle. In these cases, the energy lost is so minimal that they can be well approximated by assuming that all kinetic energy is preserved during the collision.

## Calculating Elastic Collisions

An elastic collision can be evaluated since it conserves two key quantities: momentum and kinetic energy. The below equations apply to the case of two objects that are moving with respect to each other and collide through an elastic collision.

m_{1}= Mass of object 1

m_{2}= Mass of object 2

= Initial velocity of object 1v_{1i}

= Initial velocity of object 2v_{2i}

= Final velocity of object 1v_{1f}

= Final velocity of object 2v_{2f}

Note: The boldface variables above indicate that these are the velocity vectors. Momentum is a vector quantity, so the direction matters and has to be analyzed using the tools of vector mathematics. The lack of boldface in the kinetic energy equations below is because it is a scalar quantity and, therefore, only the magnitude of the velocity matters.

Kinetic Energy of an Elastic Collision

K_{i}= Initial kinetic energy of the system

K_{f}= Final kinetic energy of the system

K_{i}= 0.5m_{1}v_{1i}^{2}+ 0.5m_{2}v_{2i}^{2}

K_{f}= 0.5m_{1}v_{1f}^{2}+ 0.5m_{2}v_{2f}^{2}

K_{i}=K_{f}

0.5m_{1}v_{1i}^{2}+ 0.5m_{2}v_{2i}^{2}= 0.5m_{1}v_{1f}^{2}+ 0.5m_{2}v_{2f}^{2}

Momentum of an Elastic Collision

P= Initial momentum of the system_{i}

P= Final momentum of the system_{f}

P=_{i}m_{1}*+v_{1i}m_{2}*v_{2i}

P=_{f}m_{1}*+v_{1f}m_{2}*v_{2f}

P=_{i}P_{f}

m_{1}*+v_{1i}m_{2}*=v_{2i}m_{1}*+v_{1f}m_{2}*v_{2f}

You are now able to analyze the system by breaking down what you know, plugging for the various variables (don't forget the direction of the vector quantities in the momentum equation!), and then solving for the unknown quantities or quantities.