# University of Wisconsin Green Bay

Two students need to build a Rube Goldberg machine for a middle school Science Olympiad competition. To start the chain of events, they plan to shoot a (2.7 g) ping pong ball horizontally out of a tube using a spring (k = 18 N/m) that has been compressed 6.5 cm. The ping pong ball will strike another ping pong ball that is suspended from a 50. cm long string. The balls will stick together (due to 0.10 g of superglue on the second ball) and swing up to strike a third ball at the top of a ramp. If the top of the ramp is to be just at the top of the swing of the ping pong balls, how high do the students need to build it?

• In this problem, you are asked to find the height of two ping pong balls after a collision. The question about height suggests that you should use energy, and the collision suggests Conservation of Momentum. So this is not a one-step problem.

If non-conservative forces are either known or small (as is the case in this problem except during the collision) and if energy is converted from one form to another between the locations, then any time you relate speed and position of an object at two different points conservation of energy is the most direct way to understand the problem. In this case, you are given one position (on a spring) and asked for another (height) with a collision in between.

This, then, is a three part problem. Energy is conserved before the collision; momentum is conserved during the collision; energy is conserved after the collision. If you do not recognize all parts of the problem before you begin, that is fine. You can start the problem with either momentum or energy and will quickly find that you need to do additional steps to find the intermediate velocities.

• For Conservation of Momentum problems, you always draw a picture of the system immediately before the collision or separation and another picture immediately after (Points B and C.) Because momentum depends on mass and velocity, label all mass and velocity information on the pictures. This helps to avoid mistakes as you fill into the equation later.

For Conservation of Energy problems, you want to show the velocity and position information at all points over which you track energy (Points A to B and Points C to D.)

• pb = pc

Any time you understand the motion of a system for which Fextermal Δt≈0, you begin with the Conservation of Momentum equation.

KEA + PEA = KEB + PEB – Wnc

KEC + PEC = KED + PED – Wnc

Any time you understand the motion of an object by looking at its energy, you begin with the Conservation of Energy equation. This form of the equation works whenever you can track Wnc.

In this problem, we know information about Point A and want to learn about Point D. Therefore, we will begin with the Conservation of Energy equation—we can use information about Point A to learn about Point B. Information about Point B will provide information about Point C through Conservation of Momentum because we do not know Wnc during that interval. The velocity found for Point C will allow us to learn about height at Point D through Conservation of Energy.

• Step One

As it leaves the spring, the ping pong ball has a speed of 5.31 m/s. Scroll down to use this information to find the speed of the two ping pong balls just after the collision.

Step Two

Following the collision, the ping pong balls swing upward with an initial speed of 2.61 m/s. Scroll down to use this information and Conservation of Energy to find out how high they rise.

Step Three

At the top of the swing, the ping pong balls reach a height of 0.35 m, so the ramp needs to be built to 0.35 m above the initial height of the ping pong balls. No further mathematical solution is necessary.

• In this problem, you are asked to compare positions of objects with a collision in between. The overall energy chain for the situation is

We solved the Conservation of Energy equation from A to B to find the velocity at Point B. Momentum was no conserved because of the force of the spring on the ball.

We solved the Conservation of Momentum equation across the collision (B to C) to find the velocity at Point C. Energy could not be tracked because of the unknown force of the collision.

We again used energy from C to D to arrive at a reasonable value for height of the ramp. Momentum was not conserved because of the action of gravity over this time period.