Linear Momentum: Collisions: Collisions in One Dimension

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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:

Two particles hit each other
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:

m₁v_1o + m₂v_2o = m₁v_1f + m₂v_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:

m₁v_1o² +

m₂v_2o² =

m₁v_1f² +

m₂v_2f²

Again, we are usually given the masses and the initial velocities of the two colliding particles, so we are given m₁,m₂,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: