What is coefficient of restitution?

Short Answer:

The coefficient of restitution is defined as the ratio of the relative velocity of separation after impact to the relative velocity of approach before impact between two colliding bodies. It measures the elasticity of a collision — that is, how much kinetic energy is conserved after the collision.

It is denoted by the symbol ‘e’ and has no unit (dimensionless). The value of  lies between 0 and 1. For a perfectly elastic collision, , and for a perfectly inelastic collision, . Real-life collisions have values of  between these two limits.

Detailed Explanation :

Coefficient of Restitution

The coefficient of restitution (e) is a key concept in the study of collisions and impact mechanics. It helps determine the type of collision and how much energy is lost when two bodies collide. The value of  gives an idea of how “bouncy” or elastic the collision is — higher values mean more elasticity and lower energy loss, while lower values mean greater deformation and energy loss.

This concept was introduced by Sir Isaac Newton and is sometimes also called Newton’s coefficient of restitution. It plays an important role in engineering mechanics, especially in analyzing impacts, vibrations, and collisions of moving parts in machines.

Definition

The coefficient of restitution is defined as:

Mathematically, if two bodies of masses  and  have initial velocities  and  before impact and velocities  and  after impact, then:

Here:

  • : Velocities of bodies before collision
  • : Velocities of bodies after collision
  • : Coefficient of restitution (dimensionless)

The numerator represents how fast the bodies separate after impact, while the denominator represents how fast they were approaching each other before impact.

Range and Meaning of e

The value of the coefficient of restitution always lies between 0 and 1, depending on the type of collision:

  1. Perfectly Elastic Collision (e = 1):
    • No loss of kinetic energy occurs.
    • The bodies rebound completely after collision.
    • Example: Collision between steel or glass balls.
  2. Perfectly Inelastic Collision (e = 0):
    • The bodies stick together after collision and move as a single unit.
    • Maximum kinetic energy is lost.
    • Example: A lump of clay hitting and sticking to another clay ball.
  3. Partially Elastic or Inelastic Collision (0 < e < 1):
    • Some kinetic energy is lost due to deformation, heat, or sound.
    • Most real-life collisions fall in this category, such as a rubber ball hitting the floor.

Thus, the value of  gives a clear indication of how elastic or inelastic a collision is.

Derivation of Formula

Consider two bodies  and  moving along the same straight line with velocities  and  before impact, and  and  after impact.

Before impact:

  • Velocity of approach:

After impact:

  • Velocity of separation:

According to Newton’s law of restitution,

If , it means no loss of kinetic energy.
If , the bodies stick together, losing all relative velocity.

This relation is always used along with the law of conservation of momentum:

By solving these two equations, we can find the final velocities ( and ) of both bodies after impact.

Physical Significance

The coefficient of restitution indicates the elastic behavior of colliding bodies.

  • When , the bodies behave like perfectly elastic materials — they store all energy temporarily as elastic potential energy and recover it fully after impact.
  • When , some energy is lost as heat, sound, or permanent deformation.
  • When , the bodies behave like plastic materials, losing all kinetic energy in deformation.

Hence,  helps engineers determine how materials will behave during collisions and how much energy will be retained or lost.

Factors Affecting the Coefficient of Restitution

  1. Material Properties:
    Hard materials like steel and glass have higher , while soft materials like rubber and clay have lower .
  2. Surface Condition:
    Smooth and polished surfaces increase , while rough or irregular surfaces decrease it.
  3. Temperature:
    High temperature can make materials softer, reducing .
  4. Impact Velocity:
    At very high velocities, deformation increases, reducing .
  5. Shape and Size of Bodies:
    The geometry of the colliding bodies also influences the elasticity of impact.

Examples

  1. A rubber ball dropped on the ground bounces back, but not to the same height. This shows .
  2. A steel ball striking a steel plate rebounds almost to the same height, showing .
  3. A clay ball hitting the floor does not bounce back, so .
  4. Collision of billiard balls or glass marbles —  is close to 1.
  5. Car crash —  is small due to large energy loss in deformation.

Applications in Engineering

  1. Impact and Collision Studies: Used in mechanics to calculate post-impact velocities and energy loss.
  2. Vehicle Crash Analysis: Helps in designing safer cars by controlling energy absorption during impact.
  3. Material Testing: Used to determine the elastic and damping properties of materials.
  4. Sports Engineering: Used to analyze how balls bounce (like cricket, tennis, or golf).
  5. Machine Design: Applied in analyzing impact between gears, cams, and mechanical linkages.
Conclusion

The coefficient of restitution is the ratio of the relative velocity of separation to the relative velocity of approach between two colliding bodies. It measures how elastic or inelastic a collision is and ranges from 0 to 1. In a perfectly elastic collision, , and in a perfectly inelastic collision, . The coefficient of restitution is a crucial concept in mechanical engineering for studying impacts, energy losses, and material behavior during collisions, helping in the design and safety of dynamic systems.