Agirl and her bicycle have a total mass of 40.0 kg. at the top of the hill her speed is 5.0 m/s, and her speed doubles as she rides down the hill. the hill is 10.0 m high and 100 m long. how much kinetic energy and potential energy is lost to friction?
a. 2 420 j
b. 1 500 j
c. 2 000 j
d. 3 920 j

The total energy is the bike's kinetic and gravitational potential energy. E = K+U. Conservation of energy says that the energy at the top of the hill must equal the energy at the bottom of the hill.  

So... E = K + U. By the way, K = 1/2 m v^2 and U = mgh.  

At the top she is moving (thus has kinetic energy) AND is above the ground she is on a hill (thus she has gravitational potential energy with respect to the bottom of the hill).  

So at the top her kinetic energy is 1/2 * 40 kg * (5 m/s)^2 = 500 J and her potential energy = 40 kg * 9.8 m/s/s * 10 m = 3920 J. Thus at the top her TOTAL energy is 4,420 J.  

Now let's look at the bottom. Since she's at the bottom, there's no potential energy. So the total energy is just her kinetic energy. She is now going twice as fast (10 m/s) so K = 1/2 * 40 kg* (10 m/s)^2 = 2000 J. At the bottom, her TOTAL energy is 2,000 J.  

WAIT A MINUTE! I thought energy was CONSERVED?! At the top, it's 4,420 J and at the bottom it's 2,000 J ?? Where did the extra go? Oh! It was lost due to friction in the tires. Friction is a force and, when that force is applied over a distance (the length the hill), friction does work on the bike. The work-energy theorem tells us that the amount of work friction does is the amount of energy the bike loses. In this problem, since it gave you the length of the hill, you can actually calculate the friction force (which is F = W/d = 2,420J / 100m = 24.2 N).  

But anyways... the loss of energy due to friction is just the difference between the energy at the top and the energy at the bottom.  

4,420 J - 2,000 J = 2,420 J. Answer choice A.