What is the work done by gravity on the cart from its initial position to when it reaches the bottom of the hill?
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Draw two pictures, one showing a cart at rest at the top of an incline, and another when it is rolling at the bottom of an incline. Draw velocity vectors on your sketch. Define your system. Label the distances, mass of the cart, and the kinetic energy of all objects in your system for both pictures.
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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, you need to multiply the magnitude of the force and the displacement in the same direction as the force. You can choose to use either the vertical displacement of the cart, or the distance traveled along the incline.
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Use the work-kinetic energy theorem to write an 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 your equation for the final velocity of the cart in terms of the vertical release height. (If your equation is in terms of the distance traveled along the incline, use trigonometry to relate this distance to the vertical height of the hill.)
- Does your equation depend on the steepness of the hill, as measured by the angle of the incline? If you released a car from the same height on hills with different slope steepness, will that effect how fast the cart is traveling at the bottom?
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