## Physics: Principles with Applications (7th Edition)

a. If there is no friction, the speed at the bottom of the hill does not depend on the angle of the hill. Gravity is the only force doing work on the sled, and the system is conservative. The initial gravitational potential energy of the sled (at the top, say, at height h) is fully converted into kinetic energy (at the bottom). The speed at the bottom of the hill is found in this way: $$\frac{1}{2}mv^{2} =mgh$$ $$v = \sqrt{2gh}$$ b. If there is friction, the speed at the bottom of the hill will depend on the angle of incline, because friction is nonconservative, and the amount of work it does depends on the angle of the hill. Only a fraction of the initial gravitational potential energy of the sled (at the top, say, at height h) is converted into kinetic energy (at the bottom). A more detailed analysis shows that the final speed is larger, for larger angles of incline. To take an extreme limit, if the hill is vertical, there is no normal force, the work done by friction is zero, and the final speed is maximum.