The stopping distance of a vehicle is an important safety factor. Assuming a constant braking force, use the work-energy theorem to show that a vehicle's stopping distance is proportional to the square of its initial speed. If an automobile traveling at \(45 \mathrm{~km} / \mathrm{h}\) is brought to a stop in \(50 \mathrm{~m}\), what would be the stopping distance for an initial speed of \(90 \mathrm{~km} / \mathrm{h}\)?