Sheila takes Josie, her niece, to the park to play on the slides. Josie starts from rest atop a 4.01-meter high slide. She is moving at 7.31 m/s when she reaches the bottom. What percentage of her total mechanical energy at the top of the slide is dissipated to heat as a result of friction?
Question
Sheila takes Josie, her niece, to the park to play on the slides. Josie starts from rest atop a 4.01-meter high slide. She is moving at 7.31 m/s when she reaches the bottom. What percentage of her total mechanical energy at the top of the slide is dissipated to heat as a result of friction?
Solution
To solve this problem, we first need to understand that the total mechanical energy of an object is the sum of its kinetic and potential energy.
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Calculate the initial potential energy (PE) at the top of the slide using the formula PE = mgh, where m is the mass (which we don't know yet, but will cancel out later), g is the acceleration due to gravity (9.8 m/s²), and h is the height (4.01 m).
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Calculate the final kinetic energy (KE) at the bottom of the slide using the formula KE = 1/2mv², where m is the mass (which we don't know yet, but will cancel out later) and v is the velocity (7.31 m/s).
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The energy lost due to friction is the difference between the initial potential energy and the final kinetic energy.
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The percentage of energy lost due to friction is the energy lost divided by the initial potential energy, multiplied by 100 to get a percentage.
Let's do the calculations:
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PE = m * 9.8 * 4.01 = 39.3m (Joules)
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KE = 1/2 * m * (7.31)² = 26.7m (Joules)
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Energy lost = PE - KE = 39.3m - 26.7m = 12.6m (Joules)
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Percentage energy lost = (Energy lost / PE) * 100 = (12.6m / 39.3m) * 100 = 32.06%
So, approximately 32.06% of Josie's total mechanical energy at the top of the slide is dissipated to heat as a result of friction.
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