Coefficient of Restitution - JEE Important Topic

VSAT 2022

Coefficient of Restitution - A Calculative Parameter of Collision

In a collision, the Coefficient of restitution is the analytical parameter which is used for the evaluation of the type of collision. The value of the coefficient of restitution ranges from 0 to 1.  A collision can be perfectly elastic, inelastic, and perfectly inelastic. When two bodies collide with each other, then their change in speed of colliding bodies before and after impact decides the parameter of coefficient of restitution.


This article has defined the coefficient of restitution for perfectly inelastic collision, coefficient of restitution for elastic collision, and coefficient of restitution for elastic bodies based upon the value of the coefficient of restitution. In this article, an attempt has been made for giving brief information (such as defining the coefficient of restitution, coefficient of restitution formula, the dimensional formula for the coefficient of restitution, etc.). 

Coefficient of Restitution and Its Derivation 

The coefficient of restitution is a parameter whose magnitude is defined as the ratio of the difference in final velocity to the difference in initial velocity between two objects after an event called a collision. It is a dimensionless quantity which means that it has no dimensions. 


The symbol by which the coefficient of restitution is defined is “e”. The value of the coefficient of restitution lies between zero and one. Based on the value of the coefficient of restitution, the type of collision will be decided. The coefficient of restitution is the ratio of the difference in final velocity to the difference in initial velocity. 


Let us consider two bodies moving with initial velocities u1 and u2 colliding with each other and after the collision, both bodies will move with the final velocities v1 and v2 respectively.   

  • The difference in final velocity will be $=v_2-v_1$

  • The difference in initial velocity will be $=u_2-u_1$

Therefore, the coefficient of restitution formula will be

  • $e=\dfrac{v_2-v_1}{u_2-u_1}$


Dimensional Formula of Coefficient of Restitution

The coefficient of restitution is the ratio of the difference in final velocity to the difference in initial velocity after the collision. As the coefficient of restitution is a dimensionless parameter, therefore it has no dimensional formula. 


Effect of Coefficient of Restitution on Collision

The coefficient of restitution has different effects on the different types of collision. The value of the coefficient of restitution ranges from 0 to 1 and that parameter decides the types of collisions. 


Case 1: When The Value of The Coefficient of Restitution is Zero. 

When the value of the coefficient of restitution is zero, then it will be a case of a perfectly inelastic collision. A perfectly inelastic collision is a special case of inelastic collision. In this type of collision, when two bodies collide with each other, then both of the bodies get to stick together and move further as a single body. 


In this case, a large change in kinetic energy is recorded. An example of perfectly inelastic collisions is dropping a ball of clay on the floor. The same information will be verified by the formula of coefficient of restitution. Mathematically, the coefficient of restitution formula is shown as;

$e=\dfrac{v_{2}-v_{1}}{u_{2}-u_{1}}$

$0=\dfrac{v_{2}-v_{1}}{u_{2}-u_{1}}$

$0=v_{2}-v_{1}$

$v_{2}=v_{1} $

It has been verified that the velocity of both objects will remain the same which means that they will move as a single object after the collision as defined by inelastic collision.


Case 2: When The Value of The Coefficient of Restitution is Lying Between 0 and 1

In such a case, this category belongs to the real-life cases where the collisions are not perfectly elastic and perfectly inelastic collisions. Therefore, when the value of the coefficient of restitution lies between 0 and 1, it refers to an inelastic collision.


In an inelastic collision, when two bodies collide with each other, then there will be some loss in the kinetic energy of the body. In this case, the body gets stuck to each other, indicating that one of the bodies has lost its kinetic energy. An inelastic collision is a type of collision in which the kinetic energy is not conserved but the momentum remains conserved.


The same information will be verified by the formula of coefficient of restitution. Mathematically, the coefficient of restitution formula is shown as;

$e=\dfrac{v_{2}-v_{1}}{u_{2}-u_{1}}$


The derived result can be used for the simplification of the complex problems related to the inelastic collision category. The coefficient of restitution for the elastic bodies lies between 0 to 1.


Case 3: When The Value of The Coefficient of Restitution is One.

When the value of the coefficient of restitution is one, then this type of collision is a perfectly elastic collision. In a perfectly elastic collision, when two bodies collide with each other, then there is no change occurring in the total kinetic energy of the system. It means that there is not any transformation of energy occurring in the form of heat, noise, etc. It is a special case of elastic collision. 


The same information will be verified by the formula of coefficient of restitution. Mathematically, the coefficient of restitution formula is shown as;

$e=\dfrac{v_{2}-v_{1}}{u_{2}-u_{1}}$

$1=\dfrac{v_{2}-v_{1}}{u_{2}-u_{1}}$

$u_{2}-u_{1}=v_{2}-v_{1}$

$u_{1}+v_{1}=u_{2}+v_{2} $

The above result depicted the evidence for the elastic collision as the sum of velocities before and after the collision is the same. 


For example for those conditions when the value of the coefficient of restitution is one when one ball is hit another ball and both move with different velocities in a snooker game.  


Conclusion 

This article concludes with the conceptual aspects of the coefficient of restitution by defining the coefficient of restitution and its formula and briefly explaining the effects of the coefficient of restitution for different types of collision. It has also defined the coefficient of restitution for perfectly inelastic collision, coefficient of restitution for elastic collision, and coefficient of restitution for elastic bodies based upon the value of the coefficient of restitution.


When the value of the coefficient of restitution lies within 0 and 1, then this range of the parameter is a depiction of the coefficient of restitution for elastic bodies.

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FAQs on Coefficient of Restitution - JEE Important Topic

1. Define the coefficient of restitution? Write the coefficient of restitution formula.  

The coefficient of restitution is the ratio of the difference in the final velocity of the bodies after a collision to the difference in the initial velocity of the bodies before the collision. It is a dimensionless parameter which depicts that it has no dimensions. The formula for the coefficient of restitution is given below:

 

Let us consider two objects that move with initial velocities u1 and u2 collide with each other and after the collision, both bodies will move the final velocities v1 and v2.  Therefore, the coefficient of restitution will be determined as,

The difference in final velocity of both objects will be $v_2-v_1$.

The difference in initial velocity of both objects will be $u_2-u_1$.

Therefore, the coefficient of restitution formula will be

$e=\dfrac{v_{2}-v_{1}}{u_{2}-u_{1}}$

2. What is the effect of the coefficient of restitution for perfectly inelastic collision? 

When the type of collision is a perfectly inelastic collision, then the value of the coefficient of restitution will be zero. It depicts that when the value of e is zero, the final velocity of both bodies after the collision will become equal to each other in the formula of the coefficient of restitution. When the velocity of both bodies is the same, then it means that both of them are moving further as a single body after the collision happened.  An example of perfectly inelastic collisions is dropping a ball of clay on the floor.

3. What are the affecting factors of the coefficient of restitution? How does the material of the body affect the coefficient of restitution?

There are three major factors which affect the coefficient of restitution. Those major affecting factors are the material properties of the colliding bodies, the geometry of the body and the impact velocity. 


As the material of the colliding body is harder, then it will be more elastic in nature. Since the material of the colliding body is elastic in nature then it will regain its shape. It shows that it is an elastic collision. Hence, the coefficient of restitution will be one. If the material of the body is soft and it doesn’t regain its shape, then it will be an inelastic collision. Therefore, the coefficient of restitution will be zero.

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