Collisions in Two Dimensions

Some objects, of course, do not stick together, so that less of the internal kinetic energy is removed. This is what happens in automobile accidents. This can also be identified by looking at a collision between two objects, in which one has a spring attached. When the two objects collide, the potential energy in the spring is “released”, resulting in both objects leaving the collision at a higher speed than before. Figure 42 shows this situation:

Figure 42.

In figure 42, the situation is frictionless so that momentum is conserved. The motion is one-dimensional. The potential energy of the compressed spring is released during the collision and is converted to internal kinetic energy. In such cases, the internal kinetic energy is increased for the system.

COLLISIONS IN TWO DIMENSIONS

This involves the collision of two objects that do not directly move in a single line. This includes things like the collision of billiard balls, which go off at an angle and often scatter. This is an approach that is not much different from studying kinematics in two dimensions and studying dynamics. The goal is to establish an appropriate coordinate system and to resolve the motion into two different components along an x and y axis. These components are solved separately. In these discussions, no rotation is

anticipated, although this will be discussed later. In order to avoid this problem, it is important to look at the scattering of masses that cannot spin.

As usual, the net force on the system will be zero so that momentum is always going to be conserved. The most basic condition is one in which one particle is at rest with the xaxis being parallel to the velocity of the incoming particle. There will be components of the momentum that happen along the x- and y-axis. In such a system, the py or momentum in the y direction will be zero and the px is the momentum of the incoming particle only (as there will be no momentum in the other particle). This is described in figure 43:

Figure 43.

In such cases, momentum is preserved so that M1 multiplied by V1 and M2 multiplied by V2 together will be the same value in both the before collision and after collision situations. The velocity components on the x axis will be related to the cosine of theta, where theta is the angle between the x axis and the direction of movement of the objects after the collision. The component of the velocity in the y direction is related to the sine of theta.

Remember that the total momentum is preserved, in which you need to consider the x component of the momentum and the y component of the momentum (as there will be

both components in play). Figure 44 describes the equations involved in the previous figure:

Figure 44.

In elastic situations where there are two colliding objects having an equal mass, the situation is similar to colliding billiards balls and in the case of many subatomic particles. The internal kinetic energy will be conserved and the momentum will be conserved. In such cases, the total kinetic energy of a system will be related to the sum of half the mass times the velocity squared, with separations of the x- and y-components based on the cosine and sine of theta or the angle of each object as it relates to the x-axis and y-axis, respectively.