A girl and her bicycle have a total mass of 46.0 kg.At the top of the hill her speed is 6.0 m/s,and her speed doubles as she rides down the hill.The hill is 15.0 m high and 105 m long.How much kinetic energy and potential energy is lost to friction?

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At the top of the hill, the total energy consists of both potential energy (due to height) and kinetic energy (due to speed). As she descends, some of this energy is converted into kinetic energy (due to increased speed), and some is lost to friction.1. Calculate initial potential energy (PE_initial) at the top:$$PE_{initial} = mgh = 46 \times 9.8 \times 15 = 6,774 \, J$$2. Calculate initial kinetic energy (KE_initial) at the top:$$KE_{initial} = \frac{1}{2}mv^2 = \frac{1}{2} \times 46 \times (6)^2 = 828 \, J$$3. Calculate total initial energy:$$E_{initial} = PE_{initial} + KE_{initial} = 6,774 + 828 = 7,602 \, J$$4. Calculate final kinetic energy (KE_final) when speed doubles (12 m/s):$$KE_{final} = \frac{1}{2}mv^2 = \frac{1}{2} \times 46 \times (12)^2 = 3,312 \, J$$5. Since she is at the bottom of the hill, potential energy (PE_final) is 0.6. Calculate total final energy:$$E_{final} = KE_{final} + PE_{final} = 3,312 + 0 = 3,312 \, J$$7. Calculate energy lost to friction:$$E_{lost} = E_{initial} - E_{final} = 7,602 - 3,312 = 4,290 \, J$$Therefore, the energy lost to friction is approximately 4.3E+3 J, which corresponds to choice A.

A Girl and Her Bicycle Have a Total Mass of 46.0