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When there is a collision between multiple objects and the final kinetic energy is different from the initial kinetic energy, it is said to be an inelastic collision. In these situations, the original kinetic energy is sometimes lost in the form of heat or sound, both of which are the results of the vibration of atoms at the point of collision. Though kinetic energy is not conserved in these collisions, momentum is still conserved and therefore the equations for momentum can be used to determine the motion of the various components of the collision.
Inelastic and Elastic Collisions in Real Life
A car crashes into a tree. The car, which was going at 80 miles per hour, instantaneously stops moving. At the same time, the impact results in a crashing noise. From a physics perspective, the car's kinetic energy changed drastically; much of the energy was lost in the form of sound (the crashing noise) and heat (which dissipates quickly). This type of collision is called "inelastic."
In contrast, a collision in which kinetic energy is conserved throughout the collision is called an elastic collision. In theory, elastic collisions involve two or more objects colliding with no loss of kinetic energy, and both objects continuing to move as they did before the collision. But of course, this doesn't really happen: any collision in the real world results in some form of sound or heat being given off, which means at least some kinetic energy is lost. For real-world purposes, though, some cases, such as two billiard balls colliding, are considered to be approximately elastic.
Perfectly Inelastic Collisions
While an inelastic collision occurs anytime that kinetic energy is lost during the collision, there is a maximum amount of kinetic energy that can be lost. In this sort of collision, called a perfectly inelastic collision, the colliding objects actually end up "stuck" together.
A classic example of this occurs when shooting a bullet into a block of wood. The effect is known as a ballistic pendulum. The bullet goes into the wood and starts the wood moving, but then "stops" within the wood. (I put "stop" in quotes because, since the bullet is now contained within the block of wood, and the wood has begun to move, the bullet is actually still moving as well, though it is not moving in relation to the wood. It has a static position inside the block of wood.) Kinetic energy is lost (mostly through the friction of the bullet heating the wood as it enters), and at the end, there's one object instead of two.
In this case, momentum is still used to figure out what has happened, but there are fewer objects after the collision than there were before the collision… because multiple objects are now stuck together. For two objects, this is the equation that would be used for a perfectly inelastic collision:
Equation for a Perfectly Inelastic Collision: