# University of Wisconsin Green Bay

You are towing a trailer of large rocks up a steep road that is inclined 120 to the horizontal. Although you are careful to drive at a steady speed of 15 m/s, when you get 2400 m along the road one of the rocks slides off the back of the trailer and rolls down the incline. How fast is the rock going when it reaches the bottom? If you find instead that the speed of the rock is 40 m/s at the bottom, what is the average drag force on the rock? Assume that the trailer is low enough to the ground that the rock does not lose any energy as it hits the ground.

• In this problem, you are asked to find the speed of a rock when you are given its initial speed and position. If non-conservative forces are either known or small 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. It is especially true in this case where some of the energy is converted into rotational motion as well as translational motion.

In this case, you start out with translational kinetic energy and stored energy due to the height of the rock and convert them into translational and rotational kinetic energies.

• Because energy problems relate motion and position, a useful picture is a picture of the motion with positions and velocities labeled. Energy is a scalar quantity and so you do not need to divide vectors into components.

• Any time you understand the motion of an object by looking at its energy, you begin with the Conservation of Energy equation.

• If the rock rolls without slipping, it will have a translational speed of 85 m/s at the bottom of the incline. Scroll down to find how much drag force is required if the block is to only have a speed of 40 m/s at the bottom of the hill.

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Step 2

A non-conservative force of 440 N acting against the rock is required if it is to only reach a speed of 40 m/s at the bottom of the hill.

• If no drag force is present, the initial kinetic and gravitational energies of the rock remain in the system and are converted into translational and rotational kinetic energies. The energy chain for this process is

translational kinetic energy + gravitational potential energy --> translational and rotational kinetic energy

If drag force is present, as in the second question, then some of the initial energy leaves the system and the energy chain is

translational kinetic energy + gravitational potential energy --> translational and rotational kinetic energy + thermal energy

This results in a slower velocity of the rock at the bottom of the hill.

It takes about 700 N or so to lift an adult human, and so this is a reasonable answer for the necessary force.