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Chapter 14: Problem 6

A 64 -kg sky diver jumped out of an airplane at an altitude of $0.90 \mathrm{km} .$ She opened her parachute after a while and eventually landed on the ground with a speed of \(5.8 \mathrm{m} / \mathrm{s} .\) How much energy was dissipated by air resistance during the jump?

Short Answer

Expert verified
Answer: Approximately 563,396 J of energy was dissipated by air resistance during the skydiver's jump.

Step by step solution

01

Calculate the initial potential energy

The potential energy at the beginning is given by the formula \(PE = mgh\), where m is the mass, g is the acceleration due to gravity (approximately \(9.8 \mathrm{m}/\mathrm{s}^2\)), and h is the initial height. We are given m = 64 kg and h = 0.90 km. First, we need to convert the altitude from kilometers to meters: \(0.90 \mathrm{km} = 900 \mathrm{m}\). \(PE = (64 \mathrm{kg})(9.8 \mathrm{m}/\mathrm{s}^2)(900 \mathrm{m})\)
02

Calculate the initial potential energy

Now, we can calculate the potential energy: \(PE = (64 \mathrm{kg})(9.8 \mathrm{m}/\mathrm{s}^2)(900 \mathrm{m}) = 564,480 \mathrm{J} \)
03

Calculate the final kinetic energy

The final kinetic energy is given by the formula \(KE = \dfrac{1}{2}mv^2\), where m is the mass and v is the final velocity (given as \(5.8 \mathrm{m}/\mathrm{s}\)). We can now calculate the final kinetic energy: \(KE = \dfrac{1}{2}(64 \mathrm{kg})(5.8 \mathrm{m}/\mathrm{s})^2\)
04

Calculate the final kinetic energy

Let's now calculate the final kinetic energy: \(KE = \dfrac{1}{2}(64 \mathrm{kg})(5.8 \mathrm{m}/\mathrm{s})^2 = 1083.52 \mathrm{J}\)
05

Calculate the energy dissipated by air resistance

To find the energy dissipated by air resistance, we can take the difference between the initial potential energy and the final kinetic energy: \(Energy \,dissipated\, by \,air\, resistance = PE - KE = 564,480 \mathrm{J} - 1083.52 \mathrm{J}\)
06

Calculate the energy dissipated by air resistance

Finally, we can find the energy dissipated by air resistance: \(Energy \,dissipated\, by \,air\, resistance = 564,480 \mathrm{J} - 1083.52 \mathrm{J} = 563,396.48 \mathrm{J}\) Approximately \(563,396 \mathrm{J}\) of energy was dissipated by air resistance during the skydiver's jump.

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