One method of solving elastic collision problems is given in this guide that you've studied previously. In the current assignment, you'll use a different approach that makes use of the concept of center of mass and eliminates much of the algebra. You'll work with some animations of elastic collisions and examine the results. This will help you learn to predict the results of these collisions. You'll need to approach these problems like you would a puzzle and look for patterns. As you do the following, enter responses in the corresponding WebAssign form.

Caution:  If, in the animations, you give the two objects velocities that prevent them from colliding, unnatural things will happen. Ignore these.

1. Begin by opening this animation of symmetric collisions. A symmetric collision is one in which the masses of the colliding objects are the same. The situation shown in the animation is the one diagrammed to the right. The goal is to learn to predict the final velocities for any pair of initial velocities. To aid in doing this, click Show Graph in order to display velocity vs. time graphs. Play the animation and compare initial and final velocities. (Ignore any small bumps and wiggles in the graphs, as these are artifacts of the graphics display.) Then try changing the initial velocities, running the animation, and watching the graphs. Here are some specific things you can try. Remember to click Reset after making any change.

   1. Give object 1 a greater velocity.

   2. Give object 2 a positive velocity.

   3. Give object 2 a negative velocity.

It shouldn't be long before you see a pattern.  Now answer item 1 in the WebAssign form.

2. For a 1-dimensional collision of two objects, the conservation of momentum equation is . We've dropped vector signs on the velocities with the understanding that the velocities can take on positive or negative values. This is sufficient to indicate direction in 1-dimensional collisions. For symmetric collisions, m₁ = m₂, so we can cancel the masses to obtain . We can do a similar thing for the conservation of kinetic energy equation, . We can cancel the masses and the factors of ½ in order to obtain . Where we're headed with this is to use the two conservation equations to come up with a relationship that explains the result you discovered in problem 1. This involves some algebra. The algebra isn't difficult, but we'll provide some guidance, because there is a technique that can help in solving a system of a linear equation and a quadratic equation.  We'll refer to the equations in this way:

Eq-1:       Eq-2: 

Do the following.

1. Square both sides of Eq-1. This will result in 3 terms on both sides of the equation. Subtract Eq-2 from the result of squaring Eq-1. This should leave a single term on each side of the equation. The result relates all four velocities to each other. Write your result in the WebAssign form.

2. Hopefully you'll see that the result of part a is consistent with your conclusion in problem 1. Make that argument in the WebAssign form.

3. The next step is to generalize to collisions in which the masses are different. For this purpose, it's helpful to use the concept of center of mass (CM). Any system of masses can be represented as concentrated in a point mass. The location of this point is the center of mass. The net, external force on the system can be thought of as acting on the CM. 

1. Begin by opening this animation. The black dot represents the CM of the system of the two objects. For the initial situation, it's located midway between the objects. The x-coordinate of the center-of-mass (X_cm) is given in the Outputs pane. 

2. Don't play the animation yet. Instead, try doubling the mass of Blue to see how that affects X_cm. Try tripling it. Set Blue to 1 kg and Red to 2 kg. At some point, you should be able to predict where X_cm will be located for any combination of Red and Blue masses.

3. With both masses set to 1 kg, play the animation. The CM has a special motion. What is it? Click Show Graph to verify your conclusion. The velocity of the CM as a function of time is the black line. Describe the motion of the CM in the WebAssign form.

4. Double Blue's mass and play the animation while watching the graph. How does the motion of the CM change? Try tripling Blue's mass. Change Blue's initial velocity to 3 m/s. How does the CM velocity change, and what does it depend on? Come up with a way to use the CM velocity to predict the final velocities of the objects. Test your method by selecting new combinations of masses and initial velocity.  Predict the final velocities before playing the animation. Describe your method in the WebAssign form.

4. Here's a test of the method you came up with in the last problem. For this test, we'll let the Red block have a non-zero initial velocity. Let the mass of Blue be 1 kg and the mass of Red be 2 kg. Let the initial velocity of Blue be 3 m/s and initial velocity of Red be 1.5 m/s. Determine the CM velocity and the final velocities of Red and Blue. Enter your results in the WebAssign form.