When people build dams – giant walls that hold entire lakes and rivers together – they have to build an overflow channel called a spillway, a flood protection measure.
A spillway could be something as simple as a path for water to flow over the dam, or more complicated, like a side channel. Sometimes there is just a big hole at the bottom of the dam (on the dry side) so the water can just gush out like a huge water cannon. This is how it works at Funil hydroelectric power station in Brazil. There is a good video showing the water coming out – it looks like a river in the air because it is a river in the air.
But the really cool physics of this spillway is that the speed of the water coming out of the hole mainly depends on the depth of the water behind the dam. Once the water leaves the tube, it essentially acts like a ball thrown at the same speed. Yes, you know what I’m going to do: I’m going to use the path of the water coming out of the weir to estimate the depth of the water in the reservoir.
There is actually a name for the relationship between flow and depth of water – it’s called Torricelli’s Law. Imagine you have a bucket full of water and poke a hole in the side near the bottom. We can use physics to find the speed of water as it flows.
Let’s start by considering the water level change during a very short time interval when the water is flowing. Here is a diagram:
Looking at the top of the bucket, the water level drops, even if it’s just a little. It doesn’t matter how much the water level drops; what interests us is the mass of this water, which I call dm. In physics, we use “d” to represent a differential amount of substance, so it could be just a small amount of water. This decrease in the water level at the top means that the water must disappear somewhere. In this case, it goes through the hole. The mass of the outgoing water should also be dmr. (You need to keep track of all the water.)
Now let’s think about this from an energetic point of view. Water is a closed system, so the total energy must be constant. There are two types of energy to think about in this case. First, there is the gravitational potential energy (Ug = mgy). It is the energy associated with the height of an object above the surface of the Earth, and it depends on the height, mass and gravitational field (g = 9.8 N / kg). The second type of energy is kinetic energy (K = (1/2) mv2). It is an energy which depends on the mass and the speed (v) of an object.