Understanding the Coefficient of Restitution (e)
In the study of collisions, the Coefficient of Restitution (often denoted by 'e') is a crucial dimensionless quantity that describes the 'bounciness' of a collision. It quantifies how much kinetic energy is conserved during an impact.
The Coefficient of Restitution (e) measures the ratio of relative speed after collision to relative speed before collision.
Imagine two balls colliding. The coefficient of restitution tells us how much of their initial relative speed is retained after they bounce off each other. A higher 'e' means a bouncier collision with more speed retained.
Mathematically, the coefficient of restitution is defined as the ratio of the magnitude of the relative velocity of separation to the magnitude of the relative velocity of approach. For two bodies 1 and 2, this is expressed as:
where:
- and are the initial velocities of bodies 1 and 2, respectively.
- and are the final velocities of bodies 1 and 2, respectively.
It's important to note that 'e' is always a non-negative value.
A coefficient of restitution of 1 signifies a perfectly elastic collision, where kinetic energy is conserved.
| Collision Type | Coefficient of Restitution (e) | Kinetic Energy Conservation |
|---|---|---|
| Perfectly Elastic | e = 1 | Conserved |
| Perfectly Inelastic | e = 0 | Not Conserved (maximum loss) |
| Inelastic | 0 < e < 1 | Not Conserved (partial loss) |
In perfectly inelastic collisions, the colliding bodies stick together after impact, resulting in the maximum possible loss of kinetic energy. In contrast, perfectly elastic collisions conserve both momentum and kinetic energy.
For JEE Physics, remember that while momentum is always conserved in any collision (provided no external forces act), kinetic energy is only conserved in perfectly elastic collisions.
Consider a ball dropped from a height . After bouncing, it reaches a maximum height . The coefficient of restitution can be related to these heights. The velocity just before impact is and the velocity just after impact is . Since the ball is falling and then rising, the relative velocity of approach is (assuming the ground is stationary) and the relative velocity of separation is . Therefore, . This means 'e' is the square root of the ratio of the rebound height to the initial height.
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e = sqrt(0.81) = 0.9
Understanding the coefficient of restitution is vital for analyzing various collision scenarios, from billiard balls to car crashes, and is a fundamental concept in mechanics for competitive exams like JEE.
Learning Resources
Provides a detailed explanation of the coefficient of restitution, its formula, and its relation to different types of collisions.
A clear explanation of the coefficient of restitution, including its definition, formula, and implications for elastic and inelastic collisions.
A video tutorial explaining the concept of the coefficient of restitution and its application in collision problems.
A forum discussion that delves into the nuances of collisions, momentum conservation, and the role of the coefficient of restitution.
An article covering key concepts in Center of Mass and Collisions for JEE, likely touching upon the coefficient of restitution.
Explains the coefficient of restitution with examples and its relationship with rebound height.
A comprehensive overview of the coefficient of restitution, including its mathematical definition, applications, and historical context.
Lecture notes and resources on one-dimensional collisions, which will cover the coefficient of restitution in detail.
A scientific paper exploring the physics behind bouncing balls, often involving the coefficient of restitution and energy loss.
A tutorial focused on collisions in physics, likely providing practice problems and explanations relevant to JEE.