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Question: If you throw a ball up in the air and it gets as high as it can go in five seconds, will it come back down in five seconds?
The short answer is no – it will take longer -, the reason is simple but the explanation is complicated. Let's discuss the question as best we can, given the limited space we have available.
If we were to assume that the effects of air on the ball's flight were negligible (in other words, we threw the ball up inside a tall vacuum chamber), then Newton's 2nd Law (NSL) applied to the ball's motion tells us that the ball would indeed take exactly 5 seconds to fall back to its point of release. In fact, from only the information that the ball took 5 seconds to reach its apex, we can calculate the following physical facts from NSL: (1) The speed of the ball when it left the thrower's hand was 49.0 m/s. If we assume that the ball was a baseball, then the thrower has a bright future as a major league pitcher because Randy Johnson in his prime threw 44.7 m/s (note: the mass of the ball does not enter the calculations for the motion in vacuum, but it does strongly impact how much force is required to throw it at a given speed); (2) The maximum height achieved by the ball was 122.6 m; (3) When the ball falls back down to the thrower's hand, it is going 49.0 m/s, the same speed at which it was thrown. This result can also be understood by considering the fact that gravity is a so-called conservative force and in the absence of dissipative mechanisms (like air resistance), the sum of kinetic and potential energies of the ball is conserved. The ball starts out with a known amount of kinetic energy and zero potential energy and when it arrives at the apex of its flight, it has zero kinetic energy and an amount of potential energy equal to its initial kinetic energy. As it returns to the thrower's hand, it again has zero potential energy and an amount of kinetic energy equal to its initial value – and thus the same speed.
But the question was directed at the motion of the ball in air, not vacuum. The presence of air makes the problem extremely difficult: we move from simple Newtonian particle mechanics to a very mathematical branch of physics called fluid dynamics. We can still use NSL to analyze the motion, but it is much more complicated. For starters, the ball no longer weighs as much in air as it does in vacuum because it experiences a buoyancy force given by Archimedes' principle. We'll not discuss this effect here, but the essence of it is due to the fact that the pressure in the lower atmosphere increases linearly with decreasing height: thus there is a greater pressure force acting on the bottom of the ball than on the top and hence it weighs less. But this is a small factor when compared with the next issue. Have you ever been in a car behind an 18-wheeler on the highway and felt the buffeting that seems to push the car side to side? The cause of this buffeting is the flow of air around the truck and the separation of this flow from either side of the rear of the truck in the form of rotating masses of air called vortices. All bluff (non-aerodynamic) bodies have some degree of separation depending on their shape and speed. Separation contributes, in addition to friction, to the drag force on the body. A ball is a bluff body. The dimples on a golf ball are there to try to minimize separation and hence lower the drag force on the ball: this allows it to travel further. Since the ball in our case travels through a range of speeds on its way up and down, the drag it experiences is a function of position in its flight; this is one reason why it is difficult to provide an explanation to the question's answer. So, the ball that takes 5 seconds to go as high as it can go in air, will take longer – because of drag effects - than 5 seconds to come back down. And when it does come back down, its terminal speed will be less than that with which it was thrown. The energetics are different in air than they are for vacuum. In the case of air, the sum of potential and kinetic energies is not conserved. The ball starts off with some initial kinetic energy, but when it reaches its apex, the ball's potential energy is less than its initial kinetic energy. And when it comes back down, its final kinetic energy is less than the potential energy it had at the top of its flight. So kinetic energy was lost both going up and coming down, and these losses are due to aerodynamic drag. This "lost" energy eventually turns up as heat energy in the air and the ball is a little bit warmer when it comes back down than when it was first thrown.
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