Exceptionally, this week's task does not come from the world of mathematics, but from physics. You want to fill a bucket with water and put it directly under the tap. Then turn on the tap. A circular jet shoots down, the bucket fills up quickly. But what happens to the water jet on its way down? Does it change its diameter?
Note: The question is not what happens to the jet in the bucket when it hits the water surface. If the beam changes before it reaches the bucket, please scroll down to find the solution!
The diameter of the water jet decreases on its way down. The simple explanation is as follows: We look at the cross section of the water jet directly at the tap and a little above the bucket. The same amount of water runs through both cross sections per second. We know that the jet has a circular cross section at the top and bottom. Now gravity comes into play: It accelerates the water as soon as it leaves the tap. The velocity of the water jet is therefore greater at the cross section at the bottom than at the cross section at the top.
However, because the water speed is higher at the bottom than at the top, the diameter of the jet at the bottom must be smaller so that the amount of water per second remains the same at the top and bottom. Physicists are talking about the flow that doesn't change, imagine a thick tube and a thin one with half the cross-sectional area. If the water flows through the thin jet twice as fast as through the thick one, the amount of water flowing through per second is the same in both hoses. Perhaps one could argue that the diameter of the jet does not change because the water density changes, air bubbles form inside or even a vacuum. But that does not happen because the air pressure acting from the outside counteracts this – and also the surface tension, which causes liquids to take on the smallest possible surface.
In his book "The Minsk Chickens and 99 Other Nice Problems", Jurij Tschernjak gives yet another particularly clear explanation: Imagine that it would drip from the tap. We look at two drops that fall down one after the other. The distance between the drops would increase the further they fall down. Because they fall faster and faster because of gravity. The time interval, on the other hand, would remain the same.We can now imagine a water jet as a collection of many drops – and then it becomes clear that the jet must become thinner.