# Here's a Brief Explanation of the Coefficient of Restitution

The coefficient of restitution is an important parameter pertaining to the collision of objects. Click here for a brief explanation regarding the same.

Satyajeet Vispute

Last Updated: Dec 18, 2018

Did You Know?

Golf club manufacturers used to make clubs that had a thin face, which provided a trampoline-like effect, allowing the ball to travel further. However, noting that this would be an unfair advantage, the USGA has since begun testing clubs for COR, and placed the upper limit at 0.83 for them.

What is the Coefficient of Restitution?

Types of Collisions

Elastic Collision

When two bodies collide with one another in an elastic type of collision, most (if not all) of the kinetic energy of the bodies, before the collision, and after the collision, remains the same.

**Example:**A good example of an almost perfectly elastic collision is when you throw a rubber ball against a wall. The ball is able to bounce back from the wall at almost the same speed that you threw it with. Therefore, as most of its kinetic energy remains the same, this collision is an elastic collision.

Inelastic Collision

The opposite of an elastic collision, an inelastic collision is one wherein most of the kinetic energy of the colliding bodies gets converted into other forms of energy during the collision. A perfectly inelastic collision occurs when all the kinetic energy in the system of the colliding bodies is converted into other forms of energy, and is lost.

**Example:**A head-on collision between two cars is a good example of an inelastic collision. Almost all the kinetic energy of the colliding cars is used up for the generation of heat due to friction, and for deforming the bodies of the vehicles. This makes it an almost perfectly inelastic collision.

Understanding the COR

How to Calculate the Coefficient of Restitution

**COR (e) = (speed of separation)/(speed of approach)**

**COR (e) = (V**

_{B}-V_{A})/(U_{B}-U_{A})Here, V

_{1}and V

_{2}are the initial and final velocities of object 1, while U

_{1}and U

_{2}are the initial and final velocities of object 2.

*The difference in the velocities of the colliding bodies after the collision is divided by the difference in their velocities before collision, to obtain the COR.*

Practice Problems

Example 1:

Particle A is traveling at 4 m/s. Particle B is traveling in the opposite direction to it at a speed of 3m/s. After the particles collide, A comes to rest. If the coefficient of restitution between the particles is 0.5, what will the speed of B be after the collision?

**Solution:**

A and B are approaching each other at a relative speed of (4 + 3) = 7 m/s

COR (e) = (speed of separation)/(speed of approach)

(speed of separation) = 7 × 0.5

speed of separation = 3.5 m/s

Since it is given that A is brought to rest after the collision, the speed of separation is equal to the speed of particle B.

**Speed of particle B = 3.5 m/s**

Example 2:

A 46-gm golf ball is struck with a golf club weighing 210 gm. The velocity of the golf club prior to impact with the ball is 50 m/s. If the coefficient of restitution between the club head and the ball is 0.8, find the speed of the ball immediately after impact.

**Solution:**

The initial speed of the ball: U

_{ball}= 0 m/s

The initial speed of club (before impact): U

_{club}= 50 m/s

M

_{ball}= 46 gm

M

_{club}= 210 gm

M

_{ball}U

_{ball}+ M

_{club}U

_{club}= M

_{ball}V

_{ball}+ M

_{club}V

_{club}--------(1)

COR (e) = ( V

_{club}- V

_{ball})/( U

_{ball}- U

_{club})

e(U

_{ball}- U

_{club})= V

_{club}- V

_{ball}

V

_{club}= (U

_{ball}- U

_{club}) + V

_{ball}

M

_{ball}U

_{ball}+ M

_{club}U

_{club}= M

_{ball}V

_{ball}+ M

_{club}× [(U

_{ball}- U

_{club}) + V

_{ball}]

(46)(0)+(210)(50)=(46)V

_{ball}+ (210) x [0.8(0-50) + V

_{ball}]

(210)(50) = V

_{ball}(46 + 210) - (210)(0.8)(50)

V

_{ball}= (210)(90)/256

**V**

_{ball}= 74 m/s