Two bodies collide each other and the collision is perfectly elastic. The co-efficient of restitution of these two bodies is
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Understanding Perfectly Elastic Collisions and Coefficient of Restitution
When two bodies collide, the collision can be classified based on whether kinetic energy is conserved during the collision. A perfectly elastic collision is a type of collision where both momentum and kinetic energy are conserved.
What is the Coefficient of Restitution?
The coefficient of restitution, denoted by '$e$', is a dimensionless quantity that describes the degree to which a collision is elastic or inelastic. It is defined as the ratio of the relative velocity of separation of two bodies after collision to the relative velocity of approach before collision.
Mathematically, the coefficient of restitution is given by:
$$e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}}$$
If body 1 has initial velocity $u_1$ and final velocity $v_1$, and body 2 has initial velocity $u_2$ and final velocity $v_2$, the relative velocity of approach is $|u_1 - u_2|$ (assuming $u_1 > u_2$ for approach) and the relative velocity of separation is $|v_2 - v_1|$ (assuming $v_2 > v_1$ for separation). So, we can write:
$$e = \frac{|v_2 - v_1|}{|u_1 - u_2|}$$
Coefficient of Restitution in Different Collision Types
- For a perfectly elastic collision, kinetic energy is conserved. This means the relative velocity of separation is equal to the relative velocity of approach. In this case, $e = 1$.
- For a perfectly inelastic collision, the two bodies stick together after collision. The relative velocity of separation is zero. In this case, $e = 0$.
- For an inelastic collision, kinetic energy is not conserved, but momentum is. The relative velocity of separation is less than the relative velocity of approach. In this case, $0 < e < 1$.
Determining 'e' for Perfectly Elastic Collision
As defined, a perfectly elastic collision conserves kinetic energy. A key property derived from the conservation laws (momentum and kinetic energy) for a perfectly elastic collision is that the relative velocity of separation between the colliding bodies is equal in magnitude to their relative velocity of approach.
Since $e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}}$, and for a perfectly elastic collision, Relative velocity of separation = Relative velocity of approach, the value of $e$ becomes:
$$e = \frac{\text{Relative velocity of approach}}{\text{Relative velocity of approach}} = 1$$
Therefore, the coefficient of restitution for a perfectly elastic collision between two bodies is 1.
Important Questions from Collisions
During inelastic collision between two bodies, which of the following quantities always remain conserved?
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