91Ó°ÊÓ

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 \(3,125,000\,\mathrm{J}\).