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Linear Momentum: Collisions

Collisions in One Dimension

Terms and Formulae

Problems

The most simple case of a collision is a one-dimensional, or head-on collision. Because of the conservation of energy and momentum we are able to predict a great deal about these collisions, and to calculate relevant quantities after the collision occurs. Before we do so, however, we must define exactly what is meant by a collision.

What is a Collision?

We all know, somewhat intuitively, the common meaning of a collision: two things hitting each other. Whether the objects are two billiard balls, two particles, or two cars, this common definition applies. The definition used in physics, however, is something more precise. In physics, a collision has two aspects:

  1. Two particles hit each other
  2. A large force is felt by each particle for a relatively short amount of time.
In this way, a collision can be seen as an event that imparts a large amount of impulse to the objects involved. In addtion, recall that impulse changes momentum.

A typical collision problem involves two particles with known initial velocities colliding; we are required to calculate the final velocity of each object. If we knew the forces acting during the collision this would be easy. Usually, however, we do not, and are forced to look for other methods of solving the problem. For instance, two balls of the same masses and initial velocities upon hitting a wall bounce back with different velocities according to the "bounciness" or elasticity of the ball. We will examine the cases in which collision problems are soluble, and make some general statements about collisions.

Elastic Collisions

A special category of collisions is called elastic collisions. Formally, an elastic condition is one in which kinetic energy is conserved. This may be difficult to grasp conceptually, so consider the following test: drop a ball from a certain height. If it hits the floor and returns to its original height, the collision between the ball and the floor is elastic. Otherwise it is inelastic. Collisions between pool balls are generally elastic; car crashes are generally inelastic.

Why are these collisions special? We know with all collisions that momentum is conserved. If two particles collide we can use the following equation:

m 1 v 1o + m 2 v 2o = m 1 v 1f + m 2 v 2f    

However, we also know that, because the collision is elastic, kinetic energy is conserved. For the same situation we can use the following equation:

m 1 v 1o 2 + m 2 v 2o 2 = m 1 v 1f 2 + m 2 v 2f 2    

Again, we are usually given the masses and the initial velocities of the two colliding particles, so we are given m 1 , m 2 , v 1o and v 2o . If we use these equations together, we now have two equations and two unknowns: v 1f and v 2f . Such a situation is always soluble, and we can always find the final velocities of two particles in an elastic collision. This is a powerful use of both conservation laws we have seen so far--the two work wonderfully to predict the outcome of elastic collisions.

Inelastic Collisions

So what if energy is not conserved? Our knowledge of such situations is more limited, since we no longer know what the kinetic energy is after the collision. However, even though kinetic energy is not conserved, momentum will always be conserved. This allows us to make some statements about inelastic collisions. Specifically, if we are given the masses of the particles, both initial velocities and one final velocity we can calculate the final velocity of the last particle through the familiar equation:

m 1 v 1o + m 2 v 2o = m 1 v 1f + m 2 v 2f    

Thus we have at least a little knowledge of inelastic collisions.

There is, however, a special case of inelastic collisions in which we can predict the outcome. Consider the case in which two particles collide, and actually physically stick together. In this case, called a completely inelastic collision we only need to solve for one final velocity, and the conservation of momentum equation is enough to predict the outcome of the collision. The two particles in a completely inelastic collision must move at the same final velocity, so our linear momentum equation becomes:

m 1 v 1o + m 2 v 2o = m 1 v f + m 2 v f    

Thus

m 1 v 1o + m 2 v 2o = Mv f    

In this equation M denotes the combined mass of the particles. Thus we can solve for completely inelastic collisions, given the initial conditions.

In studying one-dimensional collisions we are essentially applying the principle of conservation of momentum. The fact that many of these problems are soluble speaks to the importance of this principle. From our understanding of collisions in one dimension, we will move on to the two dimensional case, in which the same principles are applied, but the situations themselves become more complex.

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