An airline executive decides to economize by reducing the amount of fuel required for long-distance flights. He orders the ground crew to remove the paint from the outer surface of each plane. The paint removed from a single plane has a mass of approximately \(100 \mathrm{kg} .\) (a) If the airplane cruises at an altitude of \(12000 \mathrm{m},\) how much energy is saved in not having to lift the paint to that altitude? (b) How much energy is saved by not having to move that amount of paint from rest to a cruising speed of $250 \mathrm{m} / \mathrm{s} ?$

1. Determine the mass of the paint, the acceleration due to gravity, and the altitude. The mass of the paint is given as \(m = 100\,\mathrm{kg}\). The acceleration due to gravity is a constant, \(g = 9.8\,\mathrm{m/s^2}\). The altitude is given as \(h = 12000\,\mathrm{m}\). 2. Calculate the gravitational potential energy saved by not having to lift the paint to the altitude. To find the gravitational potential energy, we use the formula: \(PE = m \cdot g \cdot h\). Plug in the values we found in step 1 to get \(PE = (100\,\mathrm{kg}) \cdot (9.8\,\mathrm{m/s^2}) \cdot (12000\,\mathrm{m})\). 3. Evaluate the expression to find the energy saved. \(PE = 100\,\mathrm{kg} \cdot 9.8\,\mathrm{m/s^2} \cdot 12000\,\mathrm{m} = 11,760,000\,\mathrm{J}\). The energy saved in not having to lift the paint to the cruising altitude is \(11,760,000\,\mathrm{J}\).

1. Determine the mass of the paint and the cruising speed. The mass of the paint is the same as part (a), \(m = 100\,\mathrm{kg}\). The cruising speed is given as \(v = 250\,\mathrm{m/s}\). 2. Calculate the kinetic energy saved by not having to move the paint from rest to the cruising speed. To find the kinetic energy, we use the formula: \(KE = \frac{1}{2}m \cdot v^2\). Plug in the values we found in step 1 to get \(KE = \frac{1}{2}(100\,\mathrm{kg}) \cdot (250\,\mathrm{m/s})^2\). 3. Evaluate the expression to find the energy saved. \(KE = 100\,\mathrm{kg} \cdot \frac{1}{2} \cdot (250\,\mathrm{m/s})^2 = 3,125,000\,\mathrm{J}\). The energy saved by not having to move that amount of paint from rest to a cruising speed of \(250\,\mathrm{m/s}\) is \(3,125,000\,\mathrm{J}\).