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Air resistance is an important factor in problems like these. Either it is supposed to be considered or it is not, but the puzzle needs to be worded to make that clear. The simplest way is to say, "without considering air resistance..." Otherwise, working the puzzle cannot presume that no air resistance ought to be included.

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Consequently, the two puzzles above make no mention of air resistance, however, the correct solutions not only must consider air resistance, but as it turns out, air resistance is entirely what those two puzzles are all about! There would be no correct solutions without considering air resistance. In the first case, air resistance has a specific physical effect on the movement of the rain drops which is a function of the DIAMETER of the drops. Smaller drops necessarily have a smaller diameter, and larger drops have a larger diameter. As the drops of water fall, the buffeting effect of the relatively still air they're running into causes minor distortions of the shape of the drops, perhaps even causing the drops to separate into two or more smaller drops; or wayward drops might bump into other wayward drops causing them to combine into one or more larger drops. But for simplicity's sake, we ought to keep just two different sizes of drops in mind, one larger and one smaller, as the puzzle is worded that way.

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Regarding the baseball, as it ascends it encounters air resistance which has the effect of slowing down the velocity of the ball in a similar way that the force of gravity has on the ball: they are both forces of resistance to the upward motion of the baseball, but obviously, air resistance (or friction) is small compared to the greater force of gravity. It's smaller but still significant. When a major league pitcher throws a fast ball, its velocity is greatest the instant it leaves his hand and due to air friction the speed of the pitch slows down as it approaches the catcher. Likewise, the puzzle ball being thrown vertically into the air has more air resistance from the very start than it does near the top of its ascent as it slows down to zero. At the apex of its flight, the baseball's initial kinetic energy is completely exchanged for potential energy which is defined by the ball's weight times the distance upwards that it has traveled. The force of air resistance starts at its maximum and decreases to its minimum as the ball goes upward, but the force of gravity remains practically constant the whole way. To be precise it decreases slightly as the ball gets further from the earth's center but the change is too small to be significant, and therefore can be disregarded. Then on its descent, the baseball encounters the same friction in reverse order starting at zero and gradually increasing to its maximum at the bottom of its fall. But that maximum at the end is still less than the initial maximum air resistance had been against the ball when it was first thrown. The whole point is, the velocity of the ball as it returns to the elevation from which it was first thrown must be less than the initial velocity it had when it was thrown, since some amount of kinetic energy has been lost going up AND coming down, due to air resistance. In contrast, if it had been thrown in a vacuum, there would have been no air resistance, so the ball would not have lost any total energy due to its movement, consequently the time it would take to ascend would be the same as the time it would take to fall back down.

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To demonstrate the importance of air resistance, here is another puzzle that disregards it:

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A large stone is 100 times heavier than a small rock, but when dropped at the same time, they fall with the same acceleration (ignoring air resistance). Why doesn't the large stone accelerate faster? Is it because of its weight, its energy, its surface area or its inertia?

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Effectively, this puzzle could be worded to have the stone and the rock falling in a vacuum. But the effect of air resistance is so small in this case, that it really makes a minuscule contribution, and therefore for simplicity can be disregarded. Nonetheless, "ignoring air resistance" should be included to make this official. The air resistance effect on the rain drops and the baseball are far more significant due to the less density of water and ball compared to the greater density of the rocks. Generally, rocks are 3 times heavier than the same volume of water and baseball-sized rocks are about twice the weight of a baseball. Density is a function of unit mass divided by unit volume.