### Objectives

To quantitatively examine some collisions between macroscopic objects.

#### Theory

Much of our knowledge of atoms and nuclei and all of our knowledge of elementary particles is derived from observations of carefully arranged collisions. The basic sequence of a collision is shown in Fig. 1, along with a convenient set of reference axes.

The goal of a collision experiment is to find out how the particles interact during the collision by measuring the masses and velocities of the particles before and after the collision. In the case of microscopic objects, we often do not have direct access to what the particles do during the collision because the lengths and times involved are too small for observation, so there is no more direct way to obtain this information. In this experiment you will study several simple collisions between macroscopic objects to gain some understanding of how these experiments work, and the effects of various conservation laws.

In many cases conservation laws restrict, but do not totally determine, the results of a collision. For example, if the force between m_{1} and m_{2} is the only force which can change their motion, then both linear and angular momentum will be conserved in the system consisting of m1

A collision sequence showing initial and final velocities and m2. In the situation shown in the image above, where only m1 is moving before the collision, this tells us immediately that at least one of the masses must be moving after the collision to carry off the initial momentum.

More generally, we can express the conservation of linear momentum by the vector relationship

\(\vec{P1i}+\vec{P2i}=\vec{P1f}+\vec{P1f}+\vec{P2f}\)

which asserts that all components of linear momentum must be conserved regardless of what happens during the actual collision. We will work on an air table which constrains the motion to the x-y plane and reduces the frictional forces in that plane nearly to zero, so we would expect Eq. 1 to hold for our examples.

Angular momentum is a little more complicated because there are two pieces for a rigid body. The part due to translational motion is

\(\vec{L_{2}}=\vec{r} \times \vec{mv}\)

where r is a vector from the chosen, fixed, origin to the center of mass and v is the velocity of the center of mass. There is also angular momentum due to the rotation or ‘spin’ of the object about the center of mass, given by

\(\vec{L_{s}}=I\vec{w}\)

where I is the moment of inertia about the center of mass and ! ! is the angular velocity about the center of mass. The total angular momentum is then \(\vec{L}=\vec{L_{s}}=L\vec{r}\) and we can write the conservation of angular momentum in the form

\(\vec{L_{1i}}+\vec{L_{2i}}=\vec{L_{1f}}+\vec{L_{2f}}\)

If the motion is limited to two dimensions, the translational angular momentum is necessarily perpendicular to the plane of motion. On an air table, the sliders can only rotate about the vertical direction, so the spin angular momentum is also perpendicular to the plane of motion, and Eq. 4 reduces to a single scalar equation. Since only internal forces act in the plane of motion, there are no external torques to change the angular momentum of the system, and we expect Eq. 4 to hold for the collisions we will study.

Although unlikely for macroscopic objects, it may also happen that the kinetic energy remains constant during the collision, so that

\(E_{1i}+E_{2i}=E_{1f}+E_{2f}\)

As with the angular momentum, the kinetic energy of a rigid object can be thought of as the sum of a piece due to translation of the center of mass and a piece due to rotation about the center of mass,

\(E=\frac{1}{2}\;mv^{2}\;+\;\frac{1}{2}\;Iw^{2}\)

If Eq. 5 is obeyed, the collision is said to be elastic, otherwise it is inelastic. (Some people restrict the term elastic to situations where the translational energy alone is constant.) Since we do not know much about what happens when two objects hit each other, we will have to discover experimentally whether our collisions are elastic or inelastic. For a collision in two dimensions with known starting conditions there are four unknown linear velocity components and two angular speeds after the collision. Eqs. 1, 4 and 5 supply at most four restrictions on these six quantities, and in fact only three if the collision is not known to be elastic.

The final velocities must, therefore, be determined by the details of the forces acting during the collision. By measuring the final velocities for various initial conditions one can learn about the interaction force between the particles, at least in principle. During the course of the experiment you will see qualitatively how this works, and also see how the conservation laws are applied to a real collision.