When Is Coefficient Of Restitution Zero

In physics and engineering, the coefficient of restitution is an important concept used to describe how elastic or inelastic a collision is between two bodies. It helps determine whether an object bounces back, loses energy, or comes to a complete stop after impact. While most collisions fall somewhere between perfectly elastic and perfectly inelastic, a special case occurs when the coefficient of restitution equals zero. Understanding when the coefficient of restitution is zero provides insights into energy loss, material behavior, and how motion is transferred in collisions.

Understanding the Coefficient of Restitution

The coefficient of restitution (often represented by the symbole) is a ratio that measures the relative velocity of two bodies after collision compared to their relative velocity before collision. It gives a numerical value between 0 and 1

• e = 1– The collision is perfectly elastic, meaning no kinetic energy is lost and the objects bounce back fully.
• 0 < e < 1– The collision is partially inelastic, with some energy lost as heat, sound, or deformation.
• e = 0– The collision is perfectly inelastic, meaning the objects do not separate after impact but stick together, losing the maximum possible kinetic energy.

When the Coefficient of Restitution is Zero

A coefficient of restitution equal to zero occurs in perfectly inelastic collisions. In this scenario, the two colliding bodies merge or move together as a single unit after impact. Instead of bouncing away, they stay attached, and much of the kinetic energy is dissipated into heat, sound, or permanent deformation of the materials involved.

For example, if two cars collide head-on and crumple into each other without rebounding apart, the collision can be approximated as having a coefficient of restitution close to zero. Similarly, a ball of clay hitting the ground and sticking without bouncing illustrates this condition clearly.

Key Characteristics of Zero Restitution

• No rebound after the collision.
• The objects stick together or move in the same direction at the same velocity after impact.
• Maximum loss of kinetic energy compared to elastic or partially inelastic collisions.
• Momentum is still conserved, but kinetic energy is not.

Examples in Real Life

Understanding when the coefficient of restitution is zero is easier with practical examples

• Clay or putty dropping on a hard surface– The material sticks without bouncing, dissipating energy as deformation.
• Bullet embedding in a wooden block– The bullet does not bounce back but remains lodged, transferring its motion to the block.
• Vehicle crash where cars lock together– After the collision, the vehicles move together as a single mass.
• Metal stamping or forging process– The hammer and workpiece interaction involves zero restitution as energy is absorbed by deformation.

Physics Behind Zero Restitution

Whene = 0, the final relative velocity between the two bodies after collision is zero. This means

v₂ – v₁ = 0

wherev₂andv₁represent the velocities of the bodies after the collision. In other words, the objects move together with the same velocity. This condition arises because all the kinetic energy that would normally contribute to rebound is absorbed by internal forces such as plastic deformation, heat, or sound production.

Energy Considerations

Even though momentum is conserved in all types of collisions, kinetic energy is not conserved when the coefficient of restitution is zero. Instead, the system undergoes a transformation where

• Kinetic energy is converted into deformation of materials.
• Energy is released as heat due to internal friction.
• Sound waves may carry away some portion of energy.

This explains why car crashes, clay impacts, or other perfectly inelastic collisions are destructive, as large amounts of energy are absorbed into material damage.

Difference Between Zero and Non-Zero Restitution

To understand zero restitution better, it is useful to compare it with other cases

• Elastic collision (e = 1)– Energy and momentum are both conserved; objects bounce off without permanent damage.
• Partially inelastic collision (0 < e < 1)– Objects rebound slightly, but some kinetic energy is lost in the process.
• Perfectly inelastic collision (e = 0)– No rebound occurs, and the maximum kinetic energy is lost from the system.

Applications of Zero Restitution Concept

Even though a perfectly inelastic collision seems destructive, it has important applications in engineering, materials science, and safety design.

• Automobile safety design– Cars are designed to crumple in collisions, absorbing energy through deformation. This behavior mimics a zero restitution collision to protect passengers from fatal forces.
• Sports equipment– Clay balls used in experiments demonstrate energy absorption properties, helping engineers study impact resistance.
• Ballistics– Understanding how bullets embed into targets helps in forensic analysis and armor design.
• Manufacturing processes– Forging, stamping, and pressing rely on zero restitution impacts to permanently shape materials.

Factors That Lead to Zero Restitution

Several conditions cause collisions where the coefficient of restitution approaches zero

• High plasticity of materials such as clay, putty, or lead.
• High friction between colliding surfaces preventing rebound.
• Permanent deformation where materials lose their original shape.
• Collisions at lower speeds where the energy is insufficient to cause rebound but enough to cause sticking.

Role of Material Properties

Materials with low elasticity and high ductility are more likely to demonstrate zero restitution. For example, rubber typically bounces back with a high restitution value, while clay absorbs impact energy and does not rebound, making its restitution coefficient nearly zero.

Why Zero Restitution Matters

Recognizing when the coefficient of restitution is zero is critical in multiple fields. It explains why certain materials absorb impact, how collisions lead to permanent damage, and why safety systems are designed to reduce rebound forces. In sports, engineering, and everyday life, understanding this principle helps predict how objects behave in collisions and how energy is transferred during impact.

The coefficient of restitution being zero occurs in perfectly inelastic collisions where two bodies stick together after impact without rebounding. In such cases, momentum is conserved but kinetic energy is lost to deformation, heat, and sound. Examples include clay sticking to a surface, bullets embedding in wood, or cars locking together in a crash. By studying zero restitution, scientists and engineers gain valuable insights into energy dissipation, safety designs, and material behavior. Far from being just a theoretical concept, it plays a significant role in real-world applications ranging from crash testing to manufacturing processes.