8.6 Collisions of point masses in two dimensions

Learning objectives

By the end of this section, you will be able to:

• Discuss two-dimensional collisions as an extension of one-dimensional analysis.
• Define point masses.
• Derive an expression for conservation of momentum along the x -axis and y -axis.
• Describe elastic collisions of two objects with equal mass.
• Determine the magnitude and direction of the final velocity given initial velocity and scattering angle.

The information presented in this section supports the following AP® learning objectives and science practices:

• 5.D.1.2 The student is able to apply the principles of conservation of momentum and restoration of kinetic energy to reconcile a situation that appears to be isolated and elastic, but in which data indicate that linear momentum and kinetic energy are not the same after the interaction, by refining a scientific question to identify interactions that have not been considered. Students will be expected to solve qualitatively and/or quantitatively for one-dimensional situations and only qualitatively in two-dimensional situations.
• 5.D.3.3 The student is able to make predictions about the velocity of the center of mass for interactions within a defined two-dimensional system.

In the previous two sections, we considered only one-dimensional collisions; during such collisions, the incoming and outgoing velocities are all along the same line. But what about collisions, such as those between billiard balls, in which objects scatter to the side? These are two-dimensional collisions, and we shall see that their study is an extension of the one-dimensional analysis already presented. The approach taken (similar to the approach in discussing two-dimensional kinematics and dynamics) is to choose a convenient coordinate system and resolve the motion into components along perpendicular axes. Resolving the motion yields a pair of one-dimensional problems to be solved simultaneously.

One complication arising in two-dimensional collisions is that the objects might rotate before or after their collision. For example, if two ice skaters hook arms as they pass by one another, they will spin in circles. We will not consider such rotation until later, and so for now we arrange things so that no rotation is possible. To avoid rotation, we consider only the scattering of point masses    —that is, structureless particles that cannot rotate or spin.

We start by assuming that ${\mathbf{F}}_{\text{net}}=0$ , so that momentum $\mathbf{p}$ is conserved. The simplest collision is one in which one of the particles is initially at rest. (See [link] .) The best choice for a coordinate system is one with an axis parallel to the velocity of the incoming particle, as shown in [link] . Because momentum is conserved, the components of momentum along the $x$ - and $y$ -axes $(p_x\text{and}{p}_y)$ will also be conserved, but with the chosen coordinate system, $p_y$ is initially zero and $p_x$ is the momentum of the incoming particle. Both facts simplify the analysis. (Even with the simplifying assumptions of point masses, one particle initially at rest, and a convenient coordinate system, we still gain new insights into nature from the analysis of two-dimensional collisions.)