What is Elastic Collision?

Newton's Cradle. Tyler Boley/Getty Images

Question: What is Elastic Collision?


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 a 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.

m1 = Mass of object 1
m2 = Mass of object 2
v1i = Initial velocity of object 1
v2i = Initial velocity of object 2
v1f = Final velocity of object 1
v2f = Final velocity of object 2

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
Ki = Initial kinetic energy of the system
Kf = Final kinetic energy of the system
Ki = 0.5m1v1i2 + 0.5m2v2i2
Kf = 0.5m1v1f2 + 0.5m2v2f2

Ki = Kf
0.5m1v1i2 + 0.5m2v2i2 = 0.5m1v1f2 + 0.5m2v2f2

Momentum of an Elastic Collision
Pi = Initial momentum of the system
Pf = Final momentum of the system
Pi = m1 * v1i + m2 * v2i
Pf = m1 * v1f + m2 * v2f

Pi = Pf
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

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.