When individuals construct dams—big partitions that maintain again whole lakes and rivers—they need to construct an overflow channel referred to as a spillway, a mitigation in opposition to flooding.
A spillway may very well be one thing so simple as a path for water to movement excessive of the dam, or extra sophisticated, like a facet channel. Sometimes, there’s simply a huge gap on the backside of the dam (on the dry facet) in order that water can simply shoot out like a huge water cannon. This is the way it works on the Funil Hydropower Plant in Brazil. There’s a nice video displaying the water popping out—it appears like a river within the air, as a result of it principally is a river within the air.
But the actually cool physics of this spillway is that the velocity of the water popping out of the opening principally simply depends upon the depth of the water behind the dam. Once the water leaves the tube, it primarily acts like a ball thrown at that very same velocity. Yes, you recognize what I’m going to do: I’m going to make use of the trajectory of the water leaving the spillway to estimate the depth of the water within the reservoir.
There’s really a title for the connection between water movement and depth—it is referred to as Torricelli’s law. Imagine you might have a bucket stuffed with water and also you poke a gap within the facet close to the underside. We can use physics to seek out the velocity of the water because it flows out.
Let’s begin by contemplating the change in water stage throughout a very quick time interval because the water drains. Here is a diagram:
Looking on the prime of the bucket, the water stage drops—even when simply a little bit. It does not actually matter how a lot the water stage decreases; what we’re inquisitive about is the mass of this water, which I label as dm. In physics, we use “d” to characterize a differential quantity of stuff, so this might simply be a tiny quantity of water. This lower in water stage on the prime signifies that the water has to go someplace. In this case, it’s leaving by way of the opening. The mass of the exiting water should even be dm. (You need to maintain monitor of all of the water.)
Now let’s consider this from an vitality perspective. The water is a closed system, so the full vitality should be fixed. There are two sorts of vitality to consider on this case. First, there’s gravitational potential vitality (Ug = mgy). This is the vitality related to the peak of an object above the floor of the Earth, and it depends upon the peak, the mass, and the gravitational subject (g = 9.eight N/kg). The second kind of vitality is kinetic vitality (Ok = (half of)mv2). This is an vitality that depends upon the mass and the velocity (v) of an object.