Coefficient of restitution
Coefficient of Restitution
The coefficient of restitution (COR) is a measure of how much kinetic energy remains after a collision between two objects. It is a dimensionless number that ranges from 0 to 1, where 0 indicates a perfectly inelastic collision (no bounce) and 1 indicates a perfectly elastic collision (no loss of kinetic energy).
Definition
The coefficient of restitution is defined as the ratio of the relative velocity of separation to the relative velocity of approach of two colliding bodies.
Mathematically, the coefficient of restitution (e) can be expressed as:
$$ e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}} $$
where:
- ( v_{1i} ) and ( v_{2i} ) are the initial velocities of objects 1 and 2, respectively, before the collision.
- ( v_{1f} ) and ( v_{2f} ) are the final velocities of objects 1 and 2, respectively, after the collision.
Types of Collisions
There are three main types of collisions, characterized by different values of the coefficient of restitution:
- Perfectly Elastic Collision: ( e = 1 )
- Partially Elastic Collision: ( 0 < e < 1 )
- Perfectly Inelastic Collision: ( e = 0 )
Conservation of Momentum
In all types of collisions, the total linear momentum is conserved, provided no external forces act on the system. The conservation of momentum can be expressed as:
$$ m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} $$
where ( m_1 ) and ( m_2 ) are the masses of the two colliding objects.
Energy Considerations
In a perfectly elastic collision, both momentum and kinetic energy are conserved. However, in partially elastic and perfectly inelastic collisions, some kinetic energy is converted into other forms of energy, such as heat or sound.
The kinetic energy before and after the collision can be calculated using:
$$ KE_{initial} = \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 $$ $$ KE_{final} = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 $$
Differences and Important Points
| Property | Perfectly Elastic | Partially Elastic | Perfectly Inelastic |
|---|---|---|---|
| Coefficient of Restitution (e) | 1 | Between 0 and 1 | 0 |
| Kinetic Energy | Conserved | Not fully conserved | Not conserved |
| Momentum | Conserved | Conserved | Conserved |
| Collision Characteristic | Objects bounce off with no loss of kinetic energy | Objects bounce off with some loss of kinetic energy | Objects stick together and move as one mass |
Examples
Example 1: Perfectly Elastic Collision
Two identical billiard balls collide head-on with equal and opposite velocities. Since the collision is perfectly elastic, ( e = 1 ), and the balls simply exchange velocities.
Example 2: Partially Elastic Collision
A tennis ball is dropped from a certain height onto a hard surface. It bounces back, but not to the original height. This indicates that the collision is partially elastic with ( 0 < e < 1 ).
Example 3: Perfectly Inelastic Collision
Two cars collide and stick together, moving as a single unit after the collision. This is an example of a perfectly inelastic collision where ( e = 0 ).
Conclusion
The coefficient of restitution is a crucial concept in understanding collisions and their outcomes. It helps in determining the post-collision velocities of objects and the amount of kinetic energy conserved or lost during the event. By using the principles of momentum conservation and the coefficient of restitution, one can analyze and predict the results of various collision scenarios.