A girl and her bicycle have a total mass of 45 kg. At the top of the hill her speed is 5.0 m/s. The hill is 15 m high and 115 m long. If the force of friction as she rides down the hill is 21 N, what is her speed at the bottom?

To solve this problem, we need to use the principles of conservation of energy and the work-energy theorem.

Step 1: Calculate the initial total energy at the top of the hill. This includes both kinetic and potential energy.

Kinetic energy (KE) = 0.5 * mass * speed^2 Potential energy (PE) = mass * gravity * height

Given: mass = 45 kg, speed = 5 m/s, height = 15 m, gravity = 9.8 m/s^2

KE = 0.5 * 45 kg * (5 m/s)^2 = 562.5 J PE = 45 kg * 9.8 m/s^2 * 15 m = 6615 J

Total initial energy = KE + PE = 562.5 J + 6615 J = 7177.5 J

Step 2: Calculate the work done against friction as the girl rides down the hill.

Work done = force * distance

Given: force = 21 N, distance = 115 m

Work done = 21 N * 115 m = 2415 J

Step 3: Calculate the final total energy at the bottom of the hill. This is the initial total energy minus the work done against friction.

Final total energy = initial total energy - work done = 7177.5 J - 2415 J = 4762.5 J

Step 4: At the bottom of the hill, all the energy is kinetic energy (since potential energy is zero), so we can set the final total energy equal to the final kinetic energy and solve for the final speed.

Final kinetic energy = 0.5 * mass * final speed^2

Solving for final speed gives:

Final speed = sqrt((2 * final kinetic energy) / mass) = sqrt((2 * 4762.5 J) / 45 kg) = 14.6 m/s

So, the girl's speed at the bottom of the hill is approximately 14.6 m/s.