Coefficient of Restitution

What is Coefficient of Restitution

Everyone’s heard of the phrase “An object in motion stays in motion” but did you know that the coefficient of restitution (COR) is what determines just how much motion an object keeps after a collision?

The coefficient of restitution (COR) is the ratio between the relative velocity of two objects before and after a collision. It represents how much kinetic energy two objects have lost or gained during a collision. COR is a dimensionless quantity that ranges between 0 and 1, with 1 indicating no loss in kinetic energy (perfectly elastic collision) and 0 indicating a complete loss in kinetic energy (perfectly inelastic collision).

A good example of the coefficient of restitution is a basketball bouncing off the ground. When a basketball hits the ground, its COR will determine how high it bounces back up into the air. If the COR is high, then the ball will bounce back up to nearly its original height; if it’s low, then the ball won’t bounce back up as high as it did originally. Other examples include pool balls hitting each other or cars colliding on a highway. In both cases, their respective CORs will determine how much kinetic energy each object loses or gains during impact.

COR is sometimes referred to as “bounciness” because it measures how well an object can bounce back after being hit by another object.

Elastic vs Inelastic Collision

Elastic and inelastic collisions are two types of collisions that describe the behavior of objects before and after they collide. The main difference between elastic and inelastic collisions is the amount of kinetic energy that is conserved during the collision.

In an elastic collision, the total kinetic energy of the objects is conserved. That is, the kinetic energy of the objects before the collision is equal to the kinetic energy of the objects after the collision. In other words, the objects bounce off each other without losing any kinetic energy to other forms of energy, such as heat or sound. Perfectly elastic collisions are characterized by a coefficient of restitution of 1.
In an inelastic collision, some of the kinetic energy is lost and converted into other forms of energy, such as heat, sound, or deformation of the objects. The kinetic energy of the objects after the collision is less than the kinetic energy before the collision. Perfectly inelastic collisions are characterized by a coefficient of restitution of 0.

Coefficient of Restitution Formula

The coefficient of restitution for a one-dimensional collision involving two objects, A and B, can be found as:

Coefficient of Restitution (e) = \frac{v_{b} - v_{a}}{u_{a} - u_{b}}

Where:

$v_{a}$ is the final speed of object A after impact
$v_{b}$ is the final speed of object B after impact
$u_{a}$ is the initial speed of object A before impact
$u_{b}$ is the initial speed of object B before impact

For an object bouncing off a stationary target, the previous equation can be reduced to:

e = \frac{v}{u}

Where:

$v$ is the final speed of the object after impact
$u$ is the initial speed of object before impact

The coefficient of restitution is a unitless value as it is the ratio between two similar quantities.

Coefficient of Restitution Example

Ball (A) has a mass of 3 kg and is moving with a velocity of 8 m/s when it makes a direct collision with ball (B), which has a mass of 2 kg and is moving with a velocity of 4 m/s. If e = 0.7, determine the velocity of each ball after the collision. Neglect the size of the balls and assume movement to the right as positive.

Diagram showing the two balls A and B before and after the collision

Solution

Step 1) From the conservation of momentum we know that:

m_{A} (u_{A}) + m_{B} (u_{B}) = m_{A} (v_{A}) + m_{B} (v_{B})

Let’s substitute the values in the previous equation:

3 (8) + 2 (-4) = 4 (v_{A}) + 3 (v_{B})

And by simlifying that we get:

3 v_{A} + 2 v_{B} = 16

Step 2) Next Let’s write down the coefficient of restitution equation:

e = \frac{v_{B} - v_{A}}{u_{A} - u_{B}}

Now let’s substitute what we have:

0.7 = \frac{v_{B} - v_{A}}{8 - (- 4)}

And by simplifying the equation we get:

8.4 = v_{B} - v_{A}

Step 4) Now we have two equations with two unknowns:

$3 v_{A} + 2 v_{B} = 16$

v_{B} - v_{A} = 8.4

By solving the two equations we find the two objects final velocities to be:

$v_{A} = - 0.16 m / s$

v_{B} = 8.24 m / s

Applications of Coefficient of Restitution

The coefficient of restitution is an important concept in physics and engineering, and has a wide range of practical applications. Some of the most common applications of the coefficient of restitution include:

Sports equipment design: The coefficient of restitution is used to design sports equipment such as balls, bats, and racquets. In particular, the coefficient of restitution is an important factor in determining the bounce of a ball, which affects how it behaves during play.
Automotive safety: The coefficient of restitution is used to study the impact of car crashes and design safety features such as airbags and crumple zones. By understanding the coefficient of restitution between different materials and components in a car, engineers can design more effective safety systems.
Materials engineering: The coefficient of restitution is used to study the behavior of materials under impact and deformation. By measuring the coefficient of restitution of different materials, engineers can design materials with specific properties for different applications.
Packaging and shipping: The coefficient of restitution is used to design and test packaging materials such as foam and bubble wrap. By understanding the coefficient of restitution of different materials, engineers can design packaging materials that can absorb shocks and protect delicate objects during shipping.

Overall, the coefficient of restitution has many practical applications in various fields. It is an important concept that helps engineers design better and more efficient systems, while also improving safety and performance.

Coefficient of Restitution Summary
Definition	The ratio of the final to initial relative velocity between two objects before and after they collide.
Symbol	$e$
Formula	$e = \frac{v_{b} - v_{a}}{u_{a} - u_{b}}$
Units	Unitless

What does the coefficient of restitution tell us?: The coefficient of restitution provides information about the elasticity of a collision between two objects. It is a measure of the ratio of the relative speed between two objects before and after a collision.

Can coefficient of restitution be negative?: The coefficient of restitution is a value that lies in the range of 0 to 1, and it can never be negative.

What is the difference between elastic and inelastic collisions?: In an elastic collision, kinetic energy is conserved, and the objects rebound without any loss of energy. In contrast, in an inelastic collision, kinetic energy is not conserved, and some of the energy is lost as heat or sound. The objects may stick together after the collision, or they may continue to move as a single object with less kinetic energy than before.

What factors affect the coefficient of restitution?: Several factors can affect the coefficient of restitution of a material, including the type of material, temperature, surface roughness, deformation, and velocity of the objects. The elasticity of the material and the energy losses due to deformation and friction during the collision also affect the coefficient of restitution.