You are driving at a constant speed on a straight highway that passes under an overpass. At time , when your car is a distance from the overpass, a prankster on the overpass releases a water balloon from rest from a height above the highway below. Answer the following questions. You should treat the car as a point particle, ignoring its height, length, and width.
(a) Suppose that you don't see the water balloon until it explodes on your windshield. At what speed were you originally traveling, given that the water balloon strikes your car? Express your answer in terms of d, h, and any relevant constants. Simplify your answer as much as possible.
(b) Now suppose that you see the prankster release the water balloon at , and you slam on your brakes at time in an attempt to avoid being hit. (Your reaction time is .) Your car slows down with a constant acceleration of magnitude. The balloon hits the highway right in front of your car at the instant your car comes to rest. At what speed were you originally traveling, given that the water balloon just barely misses your car? Express your answer in terms of d, h, a, t, and any relevant constants. Simplify your answer as much as possible.
(c) Let be the speed you found in part (a), and let be the speed you found in part (b). Which of these two speeds is larger?
Justify your answer mathematically, with comments as necessary.