## What are Elastic Collisions?

Elastic collisions are encounters between two bodies in which there is complete conservation between both momentum and kinetic energy, or the energy of motion. This type of collision is contrasts *inelastic collisions*, in which the kinetic energy transforms into a different kind of energy such as sound or heat after two bodies meet. In an elastic collision, the only form of energy that is involved is kinetic energy. The conservation of this form of energy is easily measured in elastic collisions, as kinetic energy only depends on mass and velocity. On the other hand, the conservation of energy in inelastic collisions is not easily measured, since the conversion between kinetic energy and heat or sound is unclear.

Because all sources of energy must be conserved, but not necessarily in the form of kinetic energy, it’s hard to define cases in reality that are perfectly elastic. However, because of all of the variables involved in calculating inelastic collisions, it is far easier to estimate collisions as if they were completely elastic.

## Coefficient of Restitution

Each object is a different organization of atoms. This means that different objects collide in different ways. It’s because of this that a measure of how closely they are to perfectly elastic or perfectly inelastic any object is was developed by Sir Isaac Newton.

The coefficient of restitution is a number between 0 and 1. Objects which have a coefficient of 0 are perfectly inelastic, whereas objects with a coefficient of 1 are considered perfectly elastic. It is incredibly difficult to have perfect kinetic energy conservation, and, as such, this coefficient is almost always never equal to 1. There are instances in which it can be. For example, if energy is added to the system at any point before or after the collision, then, naturally, velocity would also be added. However, situations where enough energy is added to significantly change an object’s coefficient of restitution are rare.

Since an object’s coefficient of restitution is typically in between 0 and 1, we can assume that objects closer to 1 can be approximated as an elastic collision.

The formula for the coefficient of restitution of any particular object is:

$e=differenceinvelocitiesaftercollision/differenceinvelocitiesbeforecollision=\frac{\left|{v}_{{a}_{f}}\u2013{v}_{{b}_{f}}\right|}{\left|{v}_{{a}_{i}}\u2013{v}_{{b}_{i}}\right|}$- The absolute value bars mean that only the magnitude is considered, not the sign.

## Elastic Collision Formula

Elastic collisions have a connection to the laws of momentum conservation and energy conservation. There are two equations for each law of conservation.

From the perspective of the conservation of momentum, and knowing that momentum is equal to the mass of the object multiplied by its velocity (mv), we know that, if momentum is conserved:

${m}_{{a}_{i}}{v}_{{a}_{i}}+{m}_{{b}_{i}}{v}_{{b}_{i}}={m}_{{a}_{f}}{v}_{{a}_{f}}+{m}_{{b}_{f}}{v}_{{b}_{f}}$From the perspective of the conservation of kinetic energy, and knowing that kinetic energy is equal to half of the product of mass and the squared velocity (1/2mv^{2}) , we know that, if kinetic energy is conserved:

Although two equations are not necessarily required to solve most elastic equations, if there is a case where there are two unknown variables, there are two equations that can be used.

## Elastic Collision Example

A ball with a mass of 5 kilograms (kg) is thrown with a velocity of 9 meters per second (m/s). Another ball with a mass of 5 kg is thrown in the opposite direction at the first ball with a velocity of 8 m/s. The second ball flies backward with a velocity of 7 m/s. What is the velocity of the first ball after the collision? What is the coefficient of restitution of the ball?

First, define your variables:

- ${M}_{{a}_{i}}=5kg$
- ${v}_{{a}_{i}}=9m/s$
- ${M}_{{b}_{i}}=5kg$
- ${v}_{{b}_{i}}=\u20138m/s(negativesignrepresentstheballmovingintheoppositedirection)$
- ${M}_{{a}_{f}}=5kg$
- ${v}_{{a}_{f}}=?$
- ${M}_{{b}_{f}}=5kg$
- ${v}_{{b}_{f}}=7m/s$

Next, use either the equation for the conservation of momentum or the equation for the conservation of kinetic energy to solve for the velocity of the first ball.

- ${m}_{{a}_{i}}{v}_{{a}_{i}}+{m}_{{b}_{i}}{v}_{{b}_{i}}={m}_{{a}_{f}}{v}_{{a}_{f}}+{m}_{{b}_{f}}{v}_{{b}_{f}}$
- $5*9+5*(\u20138)=5{v}_{{a}_{f}}+5*7$
- $45\u201340=5{{v}_{a}}_{f}+35$
- $5=5{v}_{{a}_{f}}+35$
- $\u201330=5{v}_{{a}_{f}}$
- ${v}_{{a}_{f}}=\u20136m/s$

In other words, due to momentum’s conservation, the second ball fully stops the first and forces it into the opposite direction at 6 m/s. Meanwhile, the initial velocity of the first ball forces the second ball back at a speed of 7 m/s.

The coefficient of restitution can be found after knowing this velocity. It is given as:

- $e=\frac{\left|{v}_{{b}_{f}}\u2013{v}_{{a}_{f}}\right|}{\left|{v}_{{b}_{i}}\u2013{v}_{{a}_{i}}\right|}$
- $e=\frac{\left|7\u20136\right|}{\left|9\u20136\right|}$
- $e=0.76$

This was closer to an elastic collision than an inelastic collision. Perfectly elastic collisions are met when the velocity of both balls after the collision is the same as their velocities before the collision. When the coefficient of restitution is between 0 and 1, it means some degree of energy is lost. According to the material the ball is made of, different final velocities can be observed. For example, when a billiard ball strikes another billiard ball, that collision is inelastic. In fact, you can hear the sound the two balls make very clearly, indicating that a lot of kinetic energy was turned into sound energy. Meanwhile, the collision of two rubber balls is not nearly as loud. If a rubber ball hit another rubber ball, the kinetic energy would be retained more effectively, resulting in a higher restitution coefficient.

## Quiz