What does the coefficient of restitution tell us about collisions?
Coefficient of Restitution is a crucial concept in JEE Main Physics, especially when analysing collisions and energy conservation. It quantifies how much relative speed two objects retain after they collide, revealing if the impact is elastic, partially elastic, or inelastic. Mastery of this topic, including the key formula, is essential to confidently solve collision numericals and understand applications like bouncing balls, vehicle safety, and sports physics.
In JEE problems and NCERT texts, the coefficient of restitution is typically symbolised as e. It is strictly defined as the ratio of the final relative velocity (after collision) to the initial relative velocity (before collision), both measured along the line of impact. Its value lies between 0 and 1 for most physical situations. The concept of collision in one or two dimensions often refers to changes in energy and momentum, with coefficient of restitution providing a direct way to classify the type of collision.
Understanding Coefficient of Restitution: Formula and Physical Meaning
The coefficient of restitution formula used in JEE Main is:
e = (Relative velocity after collision) / (Relative velocity before collision)
For two bodies with initial velocities u1 and u2 (before impact), and final velocities v1 and v2 (after), the formula becomes:
e = (v2 - v1) / (u1 - u2)
Here,
- e: coefficient of restitution (unitless)
- u1, u2: initial velocities
- v1, v2: velocities after collision
- laws of motion fully apply during collision analysis
- e = 1 → perfectly elastic collision (no kinetic energy lost)
- e = 0 → perfectly inelastic collision (maximum energy loss; bodies stick together)
- 0 < e < 1 → inelastic collision (partial energy loss in the collision)
- The value of e depends on materials and surface properties, not mass
The dimensional formula of coefficient of restitution is often queried. Since it is a ratio of two velocities, it is a dimensionless number. This is a common exam trap, so clarify this early when revising.
Types of Collisions and Coefficient of Restitution
Knowing the type of collision is vital for both theoretical and numerical JEE questions. The coefficient of restitution clarifies the distinctions. Use the table for quick revision:
| Type of Collision | e value | Key Feature |
|---|---|---|
| Perfectly elastic | 1 | No kinetic energy lost |
| Inelastic | 0 < e < 1 | Some energy lost as heat/sound |
| Perfectly inelastic | 0 | Objects stick together after collision |
A common mistake in numericals is applying the wrong direction convention for velocities. Always establish positive and negative axes before substituting values. For instance, in elastic collisions in one dimension, total kinetic energy is conserved only if e = 1.
Application of Coefficient of Restitution in JEE Main Physics
Coefficient of restitution is not only theoretical but also underpins practical problems in JEE Main Physics. Some areas and examples include:
- Analyzing projectile motion when a ball bounces off a surface
- Using conservation of momentum and restitution together to fully solve two-body collision numericals
- Studying energy loss in collision due to non-ideal surfaces or deformation
- Practical questions on rebound height: h'n+1 = e2 hn for a vertically dropped ball (perfectly vertical motion)
- Understanding differences between elastic and inelastic collisions for MCQs
A frequent pitfall is misapplying the formula when both bodies are in motion in the same direction. Mind the sign while subtracting initial and final velocities. Always refer to JEE standards for such cases, and check with previous year solved examples.
Numerical Example: Calculating Coefficient of Restitution
Let's see a classic JEE example. Two balls, A and B, move along a line. Before collision, A has u1 = +4 m/s, B has u2 = +1 m/s. After collision, A moves at v1 = +2 m/s, B at v2 = +3 m/s. Find the coefficient of restitution.
Calculation:
e = (v2 - v1) / (u1 - u2)
= (3 - 2) / (4 - 1)
= 1 / 3
So, e = 0.33 (to two decimal places). This is an inelastic collision, as expected in many real-life scenarios.
You can practice similar questions using the kinematics practice paper or related problems on Vedantu for exam readiness.
- The coefficient of restitution is always less than or equal to 1 in real-world physics.
- Higher values mean less energy loss in the collision.
- e also influences the subsequent heights of rebound in vertical bounces – a common MCQ setup.
- Misreading the direction or failing to apply the right sign is a top source of error.
Fundamental concepts from laws of motion and energy conservation are regularly mixed into coefficient of restitution calculations in comprehensive JEE problems.
For a deep dive, reinforce learning with laws of motion, momentum conservation, and work, energy and power. Interconnect these with coefficient of restitution for multi-step and advanced MCQs.
In summary, the coefficient of restitution bridges concepts of momentum, energy, and type of collision. Its application in JEE numericals is frequent and high-scoring, especially when combined with elastic and inelastic collision questions. For further guidance, Vedantu’s Physics team curates expert step-by-step solutions and revision notes tailored for JEE aspirants.
FAQs on Coefficient of Restitution: Definition, Formula, Applications & Numericals
1. What is the coefficient of restitution?
The coefficient of restitution is a numerical measure of how much kinetic energy remains for objects involved in a collision after they bounce apart. It is defined as the ratio of relative speed after collision to relative speed before collision between two objects. This value indicates how 'bouncy' a collision is and determines whether it is elastic or inelastic.
2. What does the coefficient of restitution tell us?
The coefficient of restitution reveals how effectively a system conserves kinetic energy during a collision.
- A value of 1 means the collision is perfectly elastic (no kinetic energy lost).
- A value of 0 means it is perfectly inelastic (objects stick together and maximum energy is lost).
- Values between 0 and 1 indicate partially elastic collisions.
3. What is the formula for the coefficient of restitution?
The basic coefficient of restitution formula (e) is given by:
e = (Relative speed after collision) / (Relative speed before collision)
Mathematically,
- e = (v2 - v1) / (u1 - u2)
- u1, u2: initial velocities
- v1, v2: final velocities
4. Can the coefficient of restitution be equal to 1?
Yes, the coefficient of restitution equals 1 in a perfectly elastic collision.
- All kinetic energy is conserved.
- There is no loss of energy as heat, sound, or deformation.
- It's the theoretical upper limit for e.
5. What is a high coefficient of restitution?
A high coefficient of restitution (close to 1) means that objects bounce back strongly after collision.
- Indicates minimal energy loss.
- Is typical for elastic collisions in games, sports (like golf, tennis balls), and some physical experiments.
6. What is the coefficient of restitution for a perfectly inelastic collision?
For a perfectly inelastic collision, the coefficient of restitution (e) is 0.
- Objects stick together and move with common velocity.
- Maximum kinetic energy is lost.
7. What is the coefficient of restitution for a perfectly elastic collision?
In a perfectly elastic collision, the coefficient of restitution is exactly 1.
- All kinetic energy and momentum are conserved.
- The objects rebound without any loss in speed (in the line of impact).
8. What is the coefficient of restitution formula in terms of height?
When a body drops from a height and rebounds, the coefficient of restitution (e) formula in terms of bounce heights is:
e = √(rebound height/original height)
This formula is commonly used in experiments with balls bouncing vertically.
9. What is the coefficient of restitution symbol?
The standard symbol for the coefficient of restitution is ‘e’.
- It is a dimensionless quantity.
- Commonly used in physics and engineering contexts.
10. How do you derive the coefficient of restitution?
The derivation of the coefficient of restitution is based on Newton’s law of restitution. Steps include:
- Use relative velocity along line of impact: e = (velocity of separation) / (velocity of approach)
- Substitute the initial and final velocities of the objects.
- Solve the ratio according to the direction of travel after and before collision.
11. What does a coefficient of restitution graph show?
A coefficient of restitution graph typically displays the relation between e and variables such as collision speed, height dropped, or material type. Key aspects shown include:
- How e varies with material properties (e.g. tennis ball vs. golf ball)
- Decrease in e with increased impact velocity or energy loss
- Comparisons for different objects in experiments
12. What is the coefficient of restitution of a tennis ball?
The coefficient of restitution of a tennis ball generally ranges from 0.7 to 0.85, depending on surface, temperature, and ball condition.
- Indicates a fairly elastic bounce
- Values are specified by sports standards for consistency
13. What is the significance of coefficient of restitution in golf?
In golf, the coefficient of restitution between the golf club face and ball determines how much energy is transferred to the ball.
- Higher e leads to longer ball travel (more 'springiness')
- Golf equipment is regulated to keep e within prescribed limits
