Finish the analysis of a one-dimensional elastic collision in Example 7.4. Fill in the ste...

Finish the analysis of a one-dimensional elastic collision in Example 7.4. Fill in the steps that lead to the results for the final velocities of the two balls after the collision.

Example 7.4

Elastic Collision between Balls of Different Mass

Consider again an elastic collision between two balls with one initially at rest (Fig. 7.12A), but this time assume the balls have different masses. Repeat the analysis of this collision and find the final velocities of both balls.

Figure 7.12 Elastic collision in one dimension between two billiard balls. A One of the balls initially at rest. B After the collision, the other ball is at rest.

RECOGNIZE THE PRINCIPLE

We can again apply conservation of momentum (Eq. 7.14) and conservation of kinetic energy (Eq. 7.15). Our goal is to understand how the relative masses of the particles affect the final velocities. We follow the steps outlined in the problem-solving box on analyzing a collision. Step 1: The external forces are zero, so the momentum of the colliding particles will be conserved.

SKETCH THE PROBLEM

Step 2: Figure 7.13 shows the problem. We denote the masses as m1 and m2. One of them is initially at rest, so the velocities before the collision are v1i = v0 and v2i = 0. The final velocities (denoted v1f and v2f) are unknowns that we wish to find. This information is all collected in Figure 7.13.

Figure 7.13 Example 7.4. Analyzing an elastic collision.

IDENTIFY THE RELATIONSHIPS

We continue with our problem-solving strategy. Step 3: The conservation of momentum condition gives

We are given that kinetic energy is conserved, so we have an elastic collision.

SOLVE

Step 4: Setting the initial and final kinetic energies equal, we find

We proceed by using the momentum conservation relation (Eq. 1) to solve for the final velocity of mass 2:

Next, we substitute this result into the conservation of kinetic energy expression, Equation (2):

We can now solve for the final velocity of mass 1 and then use Equation (3) to find the final velocity of mass 2. We’ll leave the algebra for an end-of-chapter question (Question 9). We find two solutions. Solution 1 is

v1f = v0 and v2f = 0

In this solution, there is no collision at all because mass 1 simply passes through 2; that is, the final velocity of mass 1 is equal to its initial velocity, while 2 is at rest before and after the collision. These results for the final velocities do conserve kinetic energy and momentum, so they certainly satisfy our conditions for conservation of total momentum and total kinetic energy. However, this particular solution is only possible if the collision force between the two balls is zero, which does not occur in a real collision. There is a second solution for the final velocities that does describe a real collision. It is

What does it mean?

These results for the final velocities of the two balls correspond to an actual collision, and we explore some of their consequences in the end-of-chapter problems. We note one prediction here, however: if m2 is very small compared with m1 then

Hence, the final velocity of a very massive particle (m1) is approximately unchanged by the collision with a very light particle (m2). When a car collides with a mosquito, the car’s velocity barely changes.