Question

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

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

Step-by-step solution

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: \(0.90 \mathrm{km} = 900 \mathrm{m}\). \(PE = (64 \mathrm{kg})(9.8 \mathrm{m}/\mathrm{s}^2)(900 \mathrm{m})\)

Calculate the initial potential energy

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

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: \(KE = \dfrac{1}{2}(64 \mathrm{kg})(5.8 \mathrm{m}/\mathrm{s})^2\)

Calculate the final kinetic energy

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

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: \(Energy \,dissipated\, by \,air\, resistance = PE - KE = 564,480 \mathrm{J} - 1083.52 \mathrm{J}\)

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.