Water rocket physics - work
This section is all about work. Water rockets work because the pressure change in the water bottle (the rocket) does some work on the water. This work forces the water out of the nozzle, flinging the mass of the water down, which creates the equal and opposite reaction of flinging the water rocket up.
Energy in physics is vital, and work relates to energy because work is a change in energy. You can think about it in terms of work that we know, like going to work in the morning with lots of energy and coming home with less energy – hopefully, the difference equates to how much work you’ve done, or you may not keep the job very long.
Newton’s first law says something will stay at rest unless a force acts on it. Of course, nothing is at rest because we’re on a planet spinning and orbiting around the Sun, so his law is that something at rest or moving in uniform motion in a straight line will stay like that unless a force acts on it.
Kinetic Energy
In a water bottle, the mass of the water is at rest and must move to get the rocket moving; remember, we’re trying to push the water out of the bottle’s opening, which is now upside down. We need this mass to gain in velocity. It turns out that multiplying the square of the velocity by half of the mass gives us the kinetic energy of the water.
Kinetic Energy = 1/2 x mass x velocity2
So when the bottle is at rest (we haven’t launched it yet), the kinetic energy of the water is zero – the velocity is zero, and zero squared is still zero. Zero multiplied by half the mass of the water is still zero, hence zero kinetic energy. Once the launch lever is pulled and the nozzle opens, the water starts to move, so after a while, the velocity increases, and the water gains kinetic energy. So there’s the difference in energy, which equals the work done.
Why is the water starting to move? One reason is an opening created in the bottle by pulling the lever. So, just by virtue of gravity, the water would eventually all spill out – that’s not going to make the rocket fly. We need to push the water out faster than gravity makes it come out. The air in the bottle does this by being pressurised – air is compressible, so with a pump, you can squish loads more air into the bottle.
Work
To get the air into the water bottle, we have to do some work with the pump – or if you don’t want any exercise, you can use an electric pump. This work changes the potential energy in the compressed air and stores it.
Once we pull the lever and the opening to the bottle opens, the compressed air starts working to get the water moving. We want to know how much work is happening because this will be the kinetic energy of the water. That will be handy to work out how the rocket will perform.
Pressure
Boyle figured out that in enclosed space, the pressure and volume of a gas have an inverse relationship. This means if you increase one, then the other one decreases. As a formula, you can write it as:
PV = constant
where P is the pressure, and V is the volume.
In the water rocket, once the lever is pulled and the water starts to leave the nozzle, the volume of the area where the compressed air is starts to increase. This increase in volume means that the pressure is going to decrease.
The Graph
In the graph below, I plotted the Pressure on the y-axis and the air volume in the bottle on the x-axis. The graph’s curve shows how the pressure decreases as the volume increases.
You can move the sliders around for pressure and volume to understand how they work and the relationship between them. As the pressure approaches 101325 pascals, it can’t push any more water out as the pressure inside the bottle is the same as the outside pressure (we’re using a standard 101325 pascals as the atmospheric pressure at sea level).
Once the air volume reaches 1000 ml, the bottle is empty of water – as the bottle is 1000 ml.
Back to Work
This graph is really awesome (if I do say so myself) as the area underneath the curve shows how much work is being done by the air pressure decreasing. This work equals the kinetic energy of the water firing out of the nozzle. The idea is to maximise the amount of work being done by the air to have maximum kinetic energy of the water. If you want to do some maths, look at the right column; otherwise, play with sliders to see how you can maximise the amount of work. The total area is equated to work on the graph and measured in Joules.
Some Maths
Why does the area underneath the graph equate to work? How do we know?
The Y axis is pressure, measured in pascals, which is Newtons per square meter – a force applied over an area. The X-axis is Volume and is measured in cubic metres, and to make the graph readable, it is converted to millilitres. If you multiply Newtons per square metre by cubic metres, you end up with a new unit called Newton metres, also known as a Joule, a measure of energy. So, if we can find the area of the graph underneath the curve, we get the total energy, and because it’s going from zero to some larger figure, it represents a change in energy, so we call it work.
Mathematically, we could approximate the area under the curve by multiplying the average pressures by the volume. If you set the graph to 25 psi and 150ml in the slider, then read the figures, you’ll get a maximum pressure of 172369 pascals and a minimum of 137485 pascals. We add these two together and divide by two to get the average of 154927 pascals. For the volume, it’s 1000 ml minus 850 ml, giving 150 ml. This is 0.00015 m3. Multiplying those together gives 23.24 J, close to the graph’s 23.09 J. We might be happy with that, or we might want to be a little more exact, calculus can help.
Calculus
The mere mention of calculus makes many people tremble in fear, but calculus can be your friend – especially if you want to accurately know the area under a curve.
In the above calculation, we approximated the area by making it a rectangle and multiplying the height by the length. We can imagine the area under the curve as being thousands of little rectangles next to each other. The height of the super thin rectangle is the pressure at that point, and the rectangle’s width is a tiny amount of volume. All we need to do to get the total area is add up all of the areas of the rectangles.
Before we do that we need to understand what the curve on the graph represents. It is depcting a function showing the relationship between pressure and volume. By integrating that function in terms of volume we can accurately determine the area under the graph and we can solve the integral between the starting and finishing volumes. By clicking the button ‘More Calculus’ below you can see how that is done.
