Coefficient of Restitution

By Deepak Yadav|Updated : October 4th, 2022

The Coefficient of Restitution is significant in constructing balls used in various sports. Understanding collision is necessary before understanding the coefficient of restitution. Any event in which two or more bodies exert forces on one other in a very brief period is referred to be a collision. The scientific use of the word "collision" has no reference to the degree of the force involved, even though the most frequent usage of the word refers to situations in which two or more objects crash with significant force.

There may be a change in the velocities of the two bodies colliding during the collision process. The coefficient of restitution is the ratio of the difference in final velocity to the difference in initial velocities of the two bodies. Let's thoroughly understand the coefficient of restitution, its formula, and its significance.

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What is the Coefficient of Restitution?

The coefficient of restitution (COR, also represented by e) is the ratio of the final to starting relative speed of two colliding objects. It typically falls between 0 and 1, with 1 denoting an elastic collision. A collision that is perfectly inelastic has a coefficient of 0, although a 0 value does not have to be perfectly inelastic. It is represented as 1000 times the COR in the Leeb rebound hardness test, but this COR is only accurate for the test and not necessarily for the evaluated material.

The value is nearly always less than 1 due to the original translational kinetic energy being converted into rotational kinetic energy, plastic deformation, and heat. Suppose there is an energy gain through a chemical reaction, a decrease in rotational energy, or another internal energy decrease that adds to the post-collision velocity during the impact. In that case, the value may be more than 1.

Coefficient of Restitution Meaning

Sir Isaac Newton developed many mathematical equations and rules that assist us in understanding what happens when two objects collide. One such fundamental principle is Newton's law of restitution, which states that the speed of two objects following a collision depends on the materials from which they are built.

Coefficient of Restitution Definition

The coefficient of restitution is a numerical representation of the nature of the colliding materials. We can learn about the collision's elasticity from the coefficient of restitution. A fully elastic collision is one in which there is no loss of overall kinetic energy. The maximum coefficient of restitution for this collision is e = 1. A perfectly inelastic collision is one in which the most kinetic energy is wasted. They have a restitution coefficient of e = 0. The majority of real-life collisions occur in the middle.

Coefficient of Restitution Formula

Several forces are at work when two objects collide, requiring numerous mathematical equations. Many of these rules were discovered and deduced by the same well-known scientist credited with numerous discoveries and derivations. The following is the Coefficient of Restitution's mathematical formula:

Coefficient of Restitution Formula

You can see from the following equation that you always divide the smaller integer by a more significant number. Thus, the restitution coefficient is always positive. Due to initial translational kinetic energy being converted to rotational kinetic energy, plastic deformation, and heat, the value is generally always less than one. However, suppose there is an energy gain through a chemical reaction, a decrease in rotational energy, or another internal energy decrease that adds to the post-collision velocity during the impact. In that case, it may be more than one.

Dimensional Formula for Coefficient of Restitution

The dimensional formula for the coefficient of restitution is M⁰L⁰T⁰.

Range of Values for e

The values of the Coefficient of Restitution (e) change for a different collision; for example, e = 0 for perfectly inelastic collision, whereas e = 1 for perfectly elastic collision. Let us see the complete description of the range of e for various types of collisions:

  • Coefficient of restitution for perfectly inelastic collision: e = 0
  • Coefficient of restitution for inelastic collision: 0 < e < 1; when this happens, there is an inelastic contact in the real world, and some kinetic energy is lost.
  • Coefficient of restitution for perfectly elastic collisione = 1; when this happens, the collision is fully elastic, with no kinetic energy lost, and the objects rebound from one another at the same relative speed as when they were approaching.

Importance of Coefficient of Restitution

The coefficient of Restitution helps calculate the ball's speed and height after colliding. For example, a basketball bounces higher than a tennis ball because it loses less energy when it strikes the ground. The collision is more elastic the higher the ball's Coefficient of Restitution value. Additionally, balls having a greater Coefficient of Restitution will rebound higher if we drop them from a specific height.

Important GATE Topics

Work Done By A ForceMotion Under Gravity
Dynamic ResistanceStatic Resistance
Ideal DiodeBettis Theorem
Work Done By A Constant ForceApplication Layer Protocols
Castiglianos TheoremPortal Frames

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FAQs about Coefficient of Restitution

  • If there is an energy gain through a chemical reaction, a decrease in rotational energy, or another internal energy decrease that adds to the post-collision velocity during the impact, the value may be more than 1.

  • We stated that the relative velocities prior to and following a collision has the same magnitude and the opposite sign in a collision that occurs in a completely elastic medium. This indicates that e = 1 is the coefficient of restitution for an elastic collision. The coefficient of restitution for inelastic collisions will be in the range of 0 and 1.

  • If the coefficient is very near to one, it indicates that very little kinetic energy was lost during the impact. If the coefficient is low (near zero), it indicates that a significant portion of the kinetic energy was converted to heat or otherwise absorbed by deformation.

  • Any collision has a coefficient of restitution that, in most cases, falls between zero and one. In other circumstances, though, it could be less than zero, indicating that the objects traveled through one another without fully interacting, and as a result, the momentum was only partially transferred [for example, a bullet traveling through a target].

  • A number between 0 and 1 that has a value that falls inside that range is the coefficient of restitution. It is never negative. If the formula is given in that way, the numerator (apart from the negative sign) represents the relative "velocity" of separation, while the denominator indicates the relative "velocity" of the approach.

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