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 P137. Conservation of Momentum Problems

Part A. Elastic Collision

 Before attempting the problem below, study the guide to Solving 1-dimensional Elastic Collisions.

In the following problem, the symbols a and b represent integers. Show your work as requested.

Problem statement: A glider of mass aM and initial velocity v1i collides elastically with an initially stationary glider of mass bM on a horizontal air track.

Determine the velocities v1f and v2f of the two gliders after the collision in terms of a, b, and v1i only. Use the following strategy:

1. Draw pictures of the states of the two blocks before and after the collision. Label the velocities. The direction of v2f must of course be the same as the initial direction of v1i. However, the direction of v1f may be right or left. What determines whether glider 1 continues moving in the same direction or reverses direction after the collision? (The answer to the question can be stated as an inequality between a and b.)

2. How do we know that momentum is conserved for the system of the two gliders? How do we know that kinetic energy is conserved?

3. Write the equations for conservation of momentum and kinetic energy for the system. Do this in general using the symbols a, b, M, v1i, v2i, v1f, v2f. That is, do not assume v2i = 0 at this point.

4. Now you can substitute v2i = 0 and divide out common factors to get the two equations in simplest form.

5. Review the guide, Solving 1-dimensional Elastic Collisions, which presents not only the physics but also the algebra for solving the conservation equations simultaneously. You've done the physics in parts a - c above. You need not do the algebra. Simply take the appropriate results from the guide and make the necessary substitution of symbols to obtain expressions for v1f and v2f in terms of a, b, and v1i only.
1. Having found the final velocities, let a = 3 and b = 4. Substitute these values into your results from part e to determine the final velocities as fractions of v1i. Then substitute into each of your equations for conservation of momentum and kinetic energy from part d to determine whether the left and right sides of the equations reduce to the same values. If they don't, you have at least one mistake in your work. Check your work step-by-step to find the mistake. Correct your work before submitting it.

Part B. Two-dimensional collision

 Before attempting the problems below, study the guide to Solving Conservation of Momentum Problems in Two Dimensions. Show your work according to the methods presented in the guide.
1. Two hockey pucks with Velcro collars are sliding toward each other on a frictionless, horizontal air hockey table. Puck 1 of mass m is sliding east at velocity v1 while puck 2 of mass 2m is sliding north at velocity v2. The pucks collide and their Velcro collars stick to each other. After the collision, the combined pucks move away from the collision site at constant velocity vf. Assume |v1| = |v2| = v. Numerical coefficients can be given either in terms of reduced fractions or in 3 significant figure decimal notation.

1. Determine the magnitude and direction of vf in terms of m and v only. Reduce your equations to simplest form.

2. Determine the percentage of the total initial kinetic energy that is lost in the collision.

2. This problem is #25 at the end of Chapter 9 with the exception that you are to find both the magnitude and direction of the final velocity. Open this applet, read the description, and run the animation. A symbolic solution is expected before numbers are substituted. Check your results by substituting them back into the conservation equations.