How To Find Coefficient Of Restitution

Understanding how to find the coefficient of restitution is essential in physics, especially when studying collisions and energy conservation. This concept measures the elasticity of a collision between two bodies and tells us how much kinetic energy remains after they bounce off each other. Whether you’re a student preparing for exams or someone exploring mechanics out of interest, knowing how to calculate the coefficient of restitution will help you analyze real-world scenarios more accurately and efficiently.

What Is the Coefficient of Restitution?

Definition

The coefficient of restitution (often abbreviated as COR or simplye) is a dimensionless number that measures how ‘bouncy’ a collision is. It represents the ratio of the final relative velocity to the initial relative velocity between two objects after and before impact, respectively. This value ranges from 0 to 1.

Interpretation of Values

• e = 1: Perfectly elastic collision. No kinetic energy is lost.
• e = 0: Perfectly inelastic collision. The objects stick together after collision.
• 0 < e < 1: Partially elastic collision. Some kinetic energy is lost.

Formula to Calculate Coefficient of Restitution

General Formula

The formula to find the coefficient of restitution is:

e = (V₂² − V₁²) / (V₁ − V₂)

Where:

• V₁ and V₂ are the initial velocities of object 1 and object 2 before the collision.
• V₁² and V₂² are the final velocities of object 1 and object 2 after the collision.

The numerator represents the relative velocity after the collision, and the denominator represents the relative velocity before the collision.

Important Notes

• All velocities should be taken along the line of impact.
• Sign convention is crucial. Velocity direction should be consistent.

Steps to Find the Coefficient of Restitution

Step 1: Identify the Objects Involved

Determine which two objects are colliding. Label them as Object 1 and Object 2 for clarity. You’ll need to know their velocities before and after the collision.

Step 2: Measure or Note Down Initial Velocities

Measure the velocity of each object before the collision. Use a motion sensor, stopwatch, or video analysis tools if you’re conducting an experiment. Make sure you assign positive or negative values depending on the direction of motion.

Step 3: Record the Final Velocities

After the collision, measure the velocities again. Ensure you maintain the same directional convention as in the initial measurement.

Step 4: Plug Into the Formula

Use the formulae = (V₂² − V₁²) / (V₁ − V₂)and substitute the values. Simplify to find the coefficient of restitution.

Example Problem

Scenario

Imagine a ball of mass 2 kg moving at 5 m/s collides with a stationary ball of mass 3 kg. After the collision, the first ball moves at 2 m/s, and the second ball moves at 3 m/s.

Solution

Given:

• V₁ = 5 m/s
• V₂ = 0 m/s
• V₁² = 2 m/s
• V₂² = 3 m/s

Substitute into the formula:

e = (3 − 2) / (5 − 0) = 1 / 5 = 0.2

This means the collision is partially elastic, and a significant amount of kinetic energy is lost.

Applications of Coefficient of Restitution

In Sports

In sports like basketball, tennis, and baseball, the coefficient of restitution affects how balls bounce or rebound off surfaces. Manufacturers often adjust materials to control this value for fairness and performance.

In Engineering and Design

Automotive engineers study COR in crash tests to understand impact behavior. It helps in designing safety systems like bumpers and airbags to absorb energy effectively.

In Physics Experiments

Students frequently perform COR experiments using balls dropped from various heights to calculate how energy is conserved or lost upon impact with a surface.

Factors That Influence COR

Material Properties

Different materials return different COR values. For example, rubber has a higher COR than metal due to its elasticity. A rubber ball bouncing off a floor usually has a COR close to 1, while clay has one near 0.

Surface Texture

Smoother surfaces tend to preserve more kinetic energy and increase COR. A rough surface might absorb more energy during impact, decreasing the COR.

Impact Speed

At very high velocities, materials may behave differently. Some might deform permanently, altering the COR. In general, higher speeds can result in lower COR due to heat and sound energy dissipation.

Temperature

Temperature can change a material’s flexibility. Warmer materials are usually more elastic and hence show a higher COR compared to colder, stiffer ones.

How to Measure COR Experimentally

Vertical Drop Method

One common method is to drop a ball from a known height and measure the height of its first bounce. The COR can be found using the formula:

e = √(h₂ / h₁)

• h₁ = initial drop height
• h₂ = rebound height

Procedure

• Drop the ball from a measured height.
• Measure the height to which it bounces back.
• Apply the formula to calculate the coefficient.

Limitations

This method assumes no air resistance and perfect vertical motion. For more accurate results, high-speed cameras or motion sensors may be used.

Relation to Energy Loss

Elastic and Inelastic Collisions

In an elastic collision, kinetic energy is conserved, and COR is 1. In inelastic collisions, energy is transformed into sound, heat, or deformation, resulting in COR less than 1.

Kinetic Energy Comparison

Even when momentum is conserved in a collision, energy might not be. Knowing the COR helps evaluate how much kinetic energy was retained or lost, which is useful in mechanical analysis.

Finding the coefficient of restitution is an essential part of analyzing how objects behave during collisions. By using the correct formula and understanding the underlying concepts, anyone can determine how elastic or inelastic a collision is. This knowledge has wide applications in physics, engineering, sports, and even daily life. Whether you use the velocity method or the drop height method, practicing these calculations will deepen your understanding of motion, energy, and material behavior during impact.