Samovar, which holds 30 cups full of water. You put a glass under the faucet and with the clock in the hands of the watch by a second arrow, how much time the glass is filled to the brim. Suppose that in a minute. Now we must ask: how much time will empty the entire samovar, if you leave the tap running?
Seemingly childish here is a simple arithmetic task: one glass comes in 0.5 minutes, then 30 cups will result in 15 minutes.
But do experience. It appears that the samovar empties not in a quarter of an hour, as you'd expect, and in half an hour.
What's the matter? Because the calculation is so simple!
Simple, but incorrect. It is impossible to think that the outflow velocity from the beginning to the end remains the same. When the first glass has flowed out from a samovar, a stream runs under less pressure as the level of water in the samovar dropped; it is clear that the second Cup will be filled to more than half a minute; the third will follow even lazier than that, and so on
The flow rate of any fluid from the hole in an open vessel is directly dependent on the height of the liquid column standing above the hole. Brilliant Torricelli, a pupil of Galileo first pointed to this relationship and expressed it in a simple formula:
v = √2gh
where v is the flow rate, g is the acceleration of gravity, a h - the height of the liquid level above the hole. From this formula it follows that the speed of the flowing stream does not depend on the density of the liquid: light alcohol and heavy mercury at the same level flow from the holes equally fast:
It would rather result: the mercury or alcohol? The liquid level in vessels of the same.
The formula shows that on the moon, where gravity is 6 times less than on Earth, would be required to fill a glass about 2.5 times longer than on Earth.
But let us return to our problem. If after the expiration of the samovar 20 glasses of the water level in it (counting from the hole faucet) decreased four times, 21 glass filled twice slower than the 1st. And if in the future, the water level will go down 9 times, filling the last of the glasses will need is already three times more time than filling first. Everyone knows how little water flows from the faucet samovar, which is almost emptied. Solving this problem by the methods of higher mathematics, it is possible to prove that the time needed to complete emptying of the vessel, with twice the period during which resulted if the same amount of liquid at the initial level.