Short Answer:
A perfectly inelastic collision is a type of collision in which two colliding bodies stick together after impact and move as a single body. In this collision, the momentum is conserved, but a maximum amount of kinetic energy is lost due to deformation, heat, or sound.
The coefficient of restitution () for a perfectly inelastic collision is zero (0), which means there is no rebound after impact. Examples include a bullet embedding itself in a wooden block or a lump of clay sticking to another object after collision.
Detailed Explanation :
Perfectly Inelastic Collision
A perfectly inelastic collision is defined as the collision between two bodies in which they stick together after impact and move with the same final velocity. It represents the extreme case of an inelastic collision, where the bodies undergo maximum deformation and no separation occurs after impact.
Although momentum is always conserved in any collision (elastic or inelastic), in a perfectly inelastic collision, there is a loss of kinetic energy, which is transformed into other forms such as heat, sound, vibration, or deformation energy.
This type of collision occurs commonly in real life because no material is perfectly rigid or elastic — all bodies deform to some extent on impact. Perfectly inelastic collision is therefore a useful model for analyzing practical collisions in mechanics and engineering.
Mathematical Representation
Let two bodies of masses and move in a straight line with velocities and before collision.
After collision, the two bodies stick together and move with a common velocity .
By the law of conservation of momentum:
Hence, the common velocity after collision is given by:
This equation shows that total momentum before and after collision remains the same.
However, kinetic energy is not conserved in perfectly inelastic collisions.
The loss of kinetic energy (ΔKE) is given by:
The loss in kinetic energy represents the amount of energy converted into heat, sound, or deformation.
Coefficient of Restitution
The coefficient of restitution (e) is the ratio of the relative velocity of separation to the relative velocity of approach before and after collision:
For a perfectly inelastic collision,
This means there is no rebound between the two bodies after impact — they remain attached and move together with a common velocity.
Characteristics of Perfectly Inelastic Collision
- Bodies Stick Together:
After the collision, the two colliding bodies move as a single combined body. - Momentum Conservation:
Total linear momentum of the system remains conserved before and after the impact. - Kinetic Energy Loss:
Maximum loss of kinetic energy occurs in this type of collision because some energy is transformed into deformation, sound, and heat. - Coefficient of Restitution (e = 0):
There is no rebound between the two bodies after impact. - Large Deformation:
The bodies undergo noticeable deformation and may even merge or embed into one another. - Direction of Motion:
After collision, both bodies move together in the same direction with a common velocity. - Realistic Collision:
Perfectly inelastic collisions represent real-world situations more accurately than perfectly elastic ones because most impacts involve energy loss.
Examples of Perfectly Inelastic Collision
- Bullet Embedding in a Wooden Block:
A bullet fired into a wooden block becomes embedded in it, and both move together after impact with a reduced velocity. - Clay Balls Sticking Together:
When two lumps of clay collide, they stick together and move as a single mass. - Car Crash:
In a severe automobile collision, the vehicles may crumple and lock together, moving as one mass after the impact. - Meteorite Hitting the Earth:
When a meteorite hits the earth, it gets embedded into the surface — a large-scale example of a perfectly inelastic collision. - Train Coupling:
When two train coaches couple and move together after the connection, it can be considered as a perfectly inelastic impact.
Energy Consideration
In a perfectly inelastic collision, total kinetic energy is not conserved, but momentum is always conserved. The kinetic energy before impact is partly converted into:
- Heat energy due to friction and deformation
- Sound energy produced during impact
- Potential energy due to elastic deformation
- Internal strain energy within materials
Hence, the total mechanical energy (kinetic + potential) decreases after impact, while the total energy of the system (including other forms) remains constant according to the law of conservation of energy.
Practical Importance in Engineering
The concept of a perfectly inelastic collision is important in mechanical and structural engineering for analyzing real-life impacts and designing systems that can absorb energy efficiently. Some examples include:
- Crash Testing and Safety Design:
Engineers use the principle to design vehicle structures that absorb maximum energy during collisions to protect passengers. - Material Testing:
Helps determine the ability of materials to absorb impact energy without fracturing. - Shock Absorbers:
Designed to dissipate kinetic energy from sudden shocks as heat or deformation. - Projectile Motion and Ballistics:
Used to study embedding impacts like bullets or missiles penetrating targets. - Structural Design:
Applied in designing components that can withstand impact forces, such as machine housings and protective barriers.
Conclusion
A perfectly inelastic collision is a type of collision where two bodies stick together after impact and move with a common velocity. In this collision, momentum is conserved, but kinetic energy is not, as a large part of it is lost due to deformation, heat, and sound. The coefficient of restitution is zero, indicating no rebound. Though perfectly inelastic collisions are theoretical extremes, they represent many real-world collisions where energy loss is significant. The concept is vital in engineering applications like crash safety, impact testing, and structural design.