Kinetic Energy

In elastic collisions the kinetic energy and momentum of the objects involved are conserved. In most areas of science the word "conserved'' means "constant.'' The kinetic energy of an object is ½ the product of its mass and the square of its velocity. The momentum is the product of the mass and velocity.

In this project we are going to answer the following question: "If a 10 kg ball is initially at rest and is hit elastically by a 5 kg ball moving at 20 m/sec, how fast are both balls moving afterwards?''

a. What is the initial momentum for the ball initially at rest? What is the initial momentum for the ball moving at 20 m/sec? If you add these two momentums together, what is the initial momentum for both balls?

b. The final momentum for a ball is given by the formula m ·v_f where m is the mass of the ball and v_f is the final velocity of the ball. Since we have two balls, let the final velocity of the first ball be denoted by v_1f and the final velocity of the second ball be denoted by v_2f. If we add the two momentums together, we get the equation

P = m₁v_1f + m₂v_2f .

Substitute the values for m₁ and m₂ into this equation. We do not know the values for v_1f and v_2f at this time, but will use a system of equations to solve for the velocities.

c. What is the initial kinetic energy for the first ball? What is the initial kinetic energy for the second ball? If you add these two kinetic energies together, what is the initial kinetic energy for the two balls?

d. The final kinetic energy for the first ball is given by the formula KE = ½m₁v²_1f and the final kinetic energy for the second ball is given by the formula KE = ½m₂v²_2f .

Substitute in the values for m₁ and m₂ and add these two formulas together to get the final kinetic energy for both balls.

e. Since momentum is conserved, the equation from part a and the equation from part b must be equal to each other. Set these two equations equal to each other for your first equation in this system. Likewise, kinetic energy is conserved, so the equation from part c and the equation from part d must be equal to each other. Set these two equations equal to each other for your second equation in this system. You should now have two equations in two unknowns, v_1f and v_2f

f. Solve the system of equations for v_1f and v_2f . Interpret your answers in terms of the balls with which we started this problem.