What is the Coefficient of Restitution?

What is the Coefficient of Restitution?

The coefficient of restitution, often abbreviated as COR or simply denoted by “e,” describes how much speed two objects retain along their line of contact after they collide. In other words, it’s the ratio of their relative velocities along the normal direction, comparing after the collision to before.

This coefficient typically falls between 0 and 1. When the COR is 1, the collision is perfectly elastic, meaning no kinetic energy is lost in the process. On the other end, a value of 0 signals a perfectly inelastic collision, though it’s worth noting that not every collision with a COR of zero fits that strict definition.

One common way to measure the COR is through the Leeb rebound hardness test, where the result is multiplied by 1,000 and reported as such. However, this number applies specifically to the conditions of the test—it doesn’t represent a universal property for the material across all situations.

In most practical scenarios, the COR is less than 1. This is because some of the system’s initial translational kinetic energy is usually converted into other forms, such as rotational motion, permanent deformation, or even heat.

Interestingly, there are rare cases where the COR can actually exceed 1. This can happen if energy is added during the collision, perhaps due to a chemical reaction, a decrease in rotational energy, or another internal energy shift that boosts the objects’ relative speed after impact.

Coefficient of restitution: A simple explanation

When two objects collide, a range of forces come into play, bringing several mathematical relationships into the picture.

Many of these foundational principles were established by Sir Isaac Newton, a figure whose contributions to science are almost impossible to overstate. He is responsible for a host of discoveries and theories that form the backbone of classical mechanics, and his insights remain central to how we understand motion and collisions.

Focusing on the collision of two bodies, Newton introduced a concept that has come to be known as the law of restitution. In essence, this law explains that the velocity with which two objects separate after impact is influenced by the materials involved in the collision.

Take a rubber ball bouncing on a flat, rigid surface as an example. It’s intuitive to expect the ball to rebound, but not with the full energy it had before hitting the ground. In reality, no collision is perfectly elastic, which means some energy is always lost.

(If the collision were perfectly elastic, the ball would return with precisely the same energy it had before impact, but such cases are idealized and rarely encountered in everyday life.)

Whenever an object is deformed during a collision, like a basketball hitting the floor, part of its initial energy is dissipated, often as heat or sound. This is why each subsequent bounce is a little lower; with every impact, some energy slips away.

In this context, the coefficient of restitution becomes a key factor. It effectively measures how well the energy is conserved during the bounce. A higher coefficient means the object rebounds more efficiently, leading to a “bouncier” outcome, just what you’d notice with a lively basketball.

Coefficient of Restitution Formula

The mathematical formula of the Coefficient of Restitution is given as follows:

Contrary to common belief, mathematics itself was not invented by Sir Isaac Newton in 1687; rather, Newton made significant contributions to mathematical physics, particularly with his formulation of the laws of motion. What is often referred to as “Newton’s experimental law” typically relates to his work on the physics of collisions.

If you look closely at the equation for the coefficient of restitution, you’ll notice that it involves dividing the smaller value by the larger one, which ensures the result is always positive. In most practical situations, this value falls below one.

That’s because, during a collision, some of the initial translational kinetic energy gets converted into other forms like rotational kinetic energy, heat, or is lost to permanent (plastic) deformation.

There are exceptional cases, though. For example, the coefficient of restitution can exceed one if the system gains extra energy during the collision, perhaps due to a chemical reaction, a decrease in rotational energy, or some other internal change that boosts the post-collision velocity.

Possible Values for the Coefficient of Restitution (e):

• e = 0: This describes a perfectly inelastic collision. Here, the colliding objects stick together, and there’s a total loss of relative motion along the line of impact.
• 0 < e < 1: This range is what we encounter in real-world, inelastic collisions. Some kinetic energy is always lost, often as heat or sound, but the objects do not stick together completely.
• e = 1: In this ideal scenario, you have a perfectly elastic collision. The objects bounce off each other with no loss in kinetic energy; they leave the collision with the same relative speed they had before.

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