To analyze the skier's descent down the hill, we will look at energy transformations, efficiency, and energy loss.
(a) What type of energy transformation has taken place?
The skier starts at rest at the top of the hill, which means she has gravitational potential energy (PE) due to her height. As she descends the hill, this potential energy is transformed into kinetic energy (KE), which is the energy of motion.
So, the energy transformation occurring is from gravitational potential energy to kinetic energy.
(b) Find the efficiency of the energy transformation.
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Calculate the gravitational potential energy (PE) at the top of the hill: \[ PE = mgh \] Where:
- \( m = 65 , \text{kg} \) (mass of the skier)
- \( g = 9.81 , \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 75 , \text{m} \) (height of the hill)
\[ PE = 65 \times 9.81 \times 75 = 48,192.75 , \text{J} \]
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Calculate the kinetic energy (KE) at the bottom of the hill: \[ KE = \frac{1}{2} mv^2 \] Where:
- \( v = 25 , \text{m/s} \) (speed at the bottom)
\[ KE = \frac{1}{2} \times 65 \times (25)^2 = \frac{1}{2} \times 65 \times 625 = 20,312.5 , \text{J} \]
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Calculate the efficiency: \[ \text{Efficiency} = \frac{\text{Useful energy output}}{\text{Total energy input}} \times 100% \] Here, the useful energy output is the kinetic energy at the bottom, and the total energy input is the gravitational potential energy at the top.
\[ \text{Efficiency} = \frac{20,312.5}{48,192.75} \times 100% \approx 42.19% \]
(c) What happened to the energy that was lost?
The energy that was lost (the difference between the potential energy at the top and the kinetic energy at the bottom) is likely dissipated in various forms, primarily:
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Friction: The skier may have experienced friction between her skis and the snow, which converts some of the mechanical energy into thermal energy (heat).
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Air Resistance: As the skier moves down the hill, air resistance can also work against her motion, converting some kinetic energy into heat energy.
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Sound Energy: Some energy may also be transformed into sound energy as the skier moves.
Overall, the lost energy primarily goes into heating the environment (friction with snow, air resistance) and sound, thereby decreasing the efficiency of the energy transformation.