Short Answer:

A perfectly elastic collision is a type of collision in which both momentum and kinetic energy are completely conserved. In such collisions, the bodies rebound from each other without any loss of energy due to heat, sound, or deformation.

During a perfectly elastic collision, the total energy before and after the impact remains the same, and the coefficient of restitution (e) is equal to 1. Examples include the collision between steel balls, glass marbles, or atoms and molecules in gases. These collisions are mostly idealized, as in reality, some energy loss always occurs.

Detailed Explanation :

Perfectly Elastic Collision

A perfectly elastic collision is a collision in which no kinetic energy is lost, and the bodies involved return to their original shapes and sizes after the impact. It is an ideal condition that rarely exists in real life but is very important in the study of mechanics and physics.

In an elastic collision, both momentum and kinetic energy are conserved. This means the total momentum and total kinetic energy of the system before collision are exactly equal to those after collision.

The concept of a perfectly elastic collision is widely used in theoretical studies, such as the motion of molecules in gases (kinetic theory of gases), atomic collisions, and ideal mechanical systems.

Definition

A collision is said to be perfectly elastic if both the total momentum and total kinetic energy of the system remain the same before and after collision.

Mathematically,

1. Conservation of Momentum:

1. Conservation of Kinetic Energy:

where,
= masses of the two bodies,
= initial velocities before collision,
= final velocities after collision.

The coefficient of restitution (e) for a perfectly elastic collision is equal to 1, which means the relative velocity of approach before collision equals the relative velocity of separation after collision:

Characteristics of Perfectly Elastic Collision

1. Kinetic Energy Conservation:
   No kinetic energy is lost; the total kinetic energy before and after collision is the same.
2. Momentum Conservation:
   The total momentum of the colliding bodies remains constant during and after collision.
3. No Permanent Deformation:
   The colliding bodies regain their original shapes and sizes after impact.
4. Coefficient of Restitution (e = 1):
   The relative velocity of approach and separation are equal.
5. No Energy Loss Due to Heat or Sound:
   All the energy is stored and released as elastic potential energy during collision, so there is no loss due to heat, sound, or internal friction.
6. Duration of Contact:
   The contact time between the bodies is extremely short, and the restoring forces act immediately.
7. Ideal Concept:
   Perfectly elastic collisions are mostly theoretical because real-world collisions always involve some energy loss due to deformation or friction.

Mathematical Derivation

Let two bodies of masses  and  move in the same straight line with velocities  and  before collision, and after collision, their velocities become  and .

From the law of conservation of momentum:

From the law of conservation of kinetic energy:

Simplifying the kinetic energy equation (2), we get:

which shows that the relative velocity of approach before collision equals the relative velocity of separation after collision.

From equations (1) and (2), the final velocities  and  can be calculated as:

 

These formulas are useful in solving problems involving elastic collisions between two bodies.

Examples of Perfectly Elastic Collisions

1. Collision Between Billiard Balls:
   When two billiard balls collide, most of the kinetic energy is conserved, making it an example close to a perfectly elastic collision.
2. Collision Between Steel Balls:
   Steel or glass balls used in physics experiments (Newton’s cradle) show nearly elastic behavior.
3. Atomic or Molecular Collisions:
   In gases, when molecules collide, the collision is perfectly elastic as there is no loss of kinetic energy.
4. Collision Between Gas Molecules:
   In the kinetic theory of gases, molecules are assumed to collide elastically to maintain constant energy.
5. Ideal Theoretical Collisions:
   Perfectly elastic collisions are often used in mathematical modeling to simplify dynamic analysis.

Comparison with Inelastic Collision

• In a perfectly elastic collision, both momentum and kinetic energy are conserved.
• In an inelastic collision, only momentum is conserved, and some kinetic energy is lost due to deformation, heat, or sound.
• When the bodies stick together after collision, the collision becomes perfectly inelastic.

Practical Applications

• Used in mechanical design to study impact forces and collision energy.
• Important in ballistics and space studies for calculating motion after impact.
• Used in kinetic theory of gases to model molecular motion.
• Helps in designing collision-resistant systems, such as safety bumpers and shock absorbers.
• Used in vibration and oscillation analysis to understand energy transfer between bodies.

Conclusion

A perfectly elastic collision is an ideal type of collision in which both kinetic energy and momentum are conserved. The colliding bodies rebound without any energy loss or permanent deformation, and the coefficient of restitution equals one. Although such collisions rarely occur in real life, they provide a fundamental basis for studying dynamics, molecular motion, and energy transfer in mechanical and physical systems. This concept helps engineers and scientists analyze impacts, design efficient systems, and understand energy conservation in motion.