Sina has been hired as a technical adviser for an upcoming western film. In the script, a wagon containing boxes of gold has been cut loose from the horses by an outlaw. The wagon starts from rest near the top of a hill. The outlaw plans to have the wagon roll down the hill, across a flat section of ground, and over a cliff face into a canyon. The outlaw stations some of his gang in the canyon to collect the gold from the demolished wagon. Little do they know, the Lone Ranger sees the outlaw's action from his lookout post near the base of the hill, and quickly races on horseback to intercept the wagon before it plummets into the canyon. The Lone Ranger must match the speed of the wagon at the base of the hill to hook a strong cord onto the wagon, and then lasso the other end to a large rock.

The director asks her to determine how the velocity of the stagecoach near the bottom of the hill depends on its initial release height up the hill, to coordinate a reasonable required speed for the Lone Ranger's interception. She decide to model the situation using a cart released from rest on an inclined track.

EQUIPMENT: a meterstick, stopwatch, wood blocks, an aluminum track, cart.

1. Draw two pictures, one showing the cart at rest at the top of the incline, and another when it is rolling at the bottom of the incline. Draw velocity vectors on the sketch. Define your system. Label the distances, mass of the cart, and the kinetic energy of all objects in the system for both pictures.

2. What is the work done by gravity on the cart from its initial position to when it reaches the bottom of the hill? Hint: remember that to calculate work, he needs to multiply the magnitude of the force and the displacement in the same direction as the force. He can choose to use either the vertical displacement of the cart, or the distance traveled along the incline.

3. Use the work-kinetic energy theorem to give the equation that relates the work done by gravity on the cart to the change in kinetic energy between its initial release and when it reaches the base of the hill. Assume energy dissipation is small enough to be neglected. Solve the equation for the final velocity of the cart in terms of the vertical release height. (If the equation is in terms of the distance traveled along the incline, use trigonometry to relate this distance to the vertical height of the hill.)

4. Does the equation depend on the steepness of the hill, as measured by the angle of the incline? If he released a car from the same height on hills with different slope steepness, will that affect how fast the cart is traveling at the bottom?

5. For a cart rolling down an inclined track, give the expression for the final velocity in terms of the initial (vertical) release height. Does the final velocity depend on the steepness (angle) of the incline?

Use the expression to sketch a graph of the final velocity versus the initial release height