Physics and Mathematics
Coefficient of Restitution or Coefficient of Resilience
1. Introduction
When two bodies collide, they deform momentarily and then regain their shapes.
During this collision, a part of their kinetic energy may be converted into other forms (heat, sound, etc.).
The Coefficient of Restitution (e) is a measure of how elastic or inelastic a collision is.
2. Definition
The Coefficient of Restitution (e) is defined as the ratio of the relative speed of separation to the relative speed of approach between two colliding bodies.
[e] [= \dfrac{\text{Relative speed after collision}}{\text{Relative speed before collision}}]
Let:
- [u_1], [u_2] = velocities of bodies A and B before collision
- [v_1], [v_2] = velocities of bodies A and B after collision
Then,
[e = \dfrac{v_2 – v_1}{u_1 – u_2}]
3. Range of Values
[0 \le e \le 1]
| Type of Collision | Description | Value of [e] |
|---|---|---|
| Perfectly Elastic | No loss of kinetic energy | [e = 1] |
| Perfectly Inelastic | Bodies stick together after collision | [e = 0] |
| Partially Elastic/Inelastic | Partial loss of energy | [0 < e < 1] |
4. Mathematical Derivation
Consider two bodies A and B moving along the same straight line before and after collision.
- Relative velocity of approach = [u_1 – u_2]
- Relative velocity of separation = [v_2 – v_1]
According to Newton’s experimental law of restitution:
[v_2 – v_1 = e(u_1 – u_2)]
👉 Mathematics Connection: This is a linear relation between pre- and post-collision velocities — directly linked to ratio and proportion concepts.
5. Units and Dimensions
- Unit: Dimensionless (it’s a ratio)
- Dimensional Formula: None (it’s a pure number)
6. Important Formulas to Remember
| Case | Formula | Remarks |
|---|---|---|
| Definition | [e = \dfrac{v_2 – v_1}{u_1 – u_2}] | Ratio form |
| If one body hits a stationary surface | [e = \dfrac{v}{u}] | [u] = velocity before impact, [v] = after impact |
| For perfectly elastic collision | [e = 1] | Kinetic energy conserved |
| For perfectly inelastic collision | [e = 0] | Bodies stick together |
| For oblique impact | [e] [= \frac{\text{velocity component after collision along normal}}{\text{velocity component before collision along normal}}] | Use perpendicular components |
7. Practical Examples
- A rubber ball rebounding off the ground.
- A car crash test.
- Steel ball hitting a steel plate.
- Clay ball hitting a wall (sticks → [e = 0]).
- Table tennis ball bounce test.
8. Experimental Determination
Let a ball be dropped from a height [h_1] and rebound to a height [h_2].
Then, the coefficient of restitution is given by:
[e = \sqrt{\dfrac{h_2}{h_1}}]
👉 Mathematics Link: This involves square roots and ratios — useful to cross-link with the topic Radicals and Powers.
9. Practice Questions (With Solutions)
Q1. A ball falls from a height of 5 m and rebounds to a height of 1.25 m. Find [e].
Solution:
[e = \sqrt{\dfrac{h_2}{h_1}}] [= \sqrt{\dfrac{1.25}{5}} = \sqrt{0.25} = 0.5]
Q2. A 2 kg ball moving at 10 m/s collides with a 1 kg stationary ball. If [e = 0.8], find their velocities after collision.
Solution:
Using conservation of momentum and restitution:
[m_1u_1 + m_2u_2] [= m_1v_1 + m_2v_2 \
v_2 – v_1] [= e(u_1 – u_2)]
[\Rightarrow 2(10) = 2v_1 + 1v_2,\quad v_2 – v_1 = 8]
Solving, [v_1 = 4\ \text{m/s}], [v_2 = 12\ \text{m/s}].
Q3. A rubber ball dropped from 1.6 m rebounds to 0.9 m. Find [e].
Solution:
[e = \sqrt{\dfrac{0.9}{1.6}} = 0.75]
Q4. For a ball dropped on a hard floor with [e = 0.6], find the ratio of kinetic energy after rebound to that before impact.
Solution:
[\text{Kinetic energy ratio} = e^2 = 0.36]
Q5. Two identical masses collide elastically. If one was at rest, find the final velocities.
Solution:
For elastic collision, [e = 1]:
[v_1 = 0,\ v_2 = u_1]
They exchange velocities.
10. Conceptual Questions
1. What does the coefficient of restitution signify?
It measures how elastic a collision is — i.e., how well kinetic energy is conserved.
2. Can the coefficient of restitution be greater than 1?
No, since it would violate energy conservation. Real materials always have [e ≤ 1].
3. Why is [e] dimensionless?
Because it’s the ratio of two velocities having the same unit.
4. What happens if [e = 0]?
The bodies stick together after collision — perfectly inelastic collision.
5. What happens if [e = 1]?
It is a perfectly elastic collision — no loss of kinetic energy.
6. Why is kinetic energy not conserved in inelastic collisions?
Because part of the energy converts to heat, sound, or deformation.
7. How can [e] be experimentally determined?
By dropping a ball from height [h_1] and measuring rebound height [h_2].
8. Why do different materials have different [e] values?
Due to variation in elasticity and internal energy loss during deformation.
9. Is [e] the same for all collisions of the same objects?
No, it depends on surface roughness, temperature, and speed of impact.
10. What does [e = 0.5] mean physically?
The relative speed after collision is half of the relative speed before collision.
11. Can [e] be negative?
No, because it’s defined as the ratio of magnitudes (always positive).
12. Does [e] depend on the direction of motion?
No, it depends only on the magnitudes of velocities along the line of impact.
13. Is energy conserved in all collisions?
Momentum is always conserved, but kinetic energy is conserved only in elastic collisions.
14. What happens when a ball bounces repeatedly?
The rebound height keeps decreasing because [e < 1].
15. How is the coefficient of restitution related to time of contact?
For elastic collisions, the contact time is smaller; for inelastic ones, longer due to deformation.
11. FAQs / Common Misconceptions
1. If momentum is conserved, does it mean energy is conserved too?
Not necessarily. Momentum is always conserved; energy may be partially converted to heat or sound.
2. Is [e] always 1 for steel balls?
No, even steel has [e < 1] due to slight energy loss in deformation.
3. Is coefficient of restitution applicable only to vertical impacts?
No, it applies to oblique impacts too, considering components along the normal direction.
4. Can two objects have the same [e] but different materials?
Yes, different materials can coincidentally produce the same rebound ratio.
5. Does [e] change with velocity of impact?
Yes, for most materials, [e] decreases slightly as impact velocity increases.
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