A good place to start when you want to send something to another planet is to get it off the planet it is currently on. In other words, our first step will be to launch our rocket, and have it reach escape velocity so that it truly has entered space!
To start us off, we have a look at the rocket engine, which we suspect is an essential component in rockets. Inside the rocket we have a combustion chamber with hot H2 gas that moves into several different gas chambers where we keep the gas at a fixed density and temperature - The temperature and density can vary greatly at the microscopic level, but on the macroscopic level we will consider uniform for simplicity. We choose H2 gas because it's light, and the lighter our rocket is the less fuel we need to launch it, and although normally the gas would be slightly more complicated, only using H2 gas simplifies our task.
How does the engine help the rocket travel upwards? You’ve probably heard of Newton’s third law, “for every force there is an equal and opposite force,” or \(\vec{F} = - \vec{F}\), but what does this mean for our rocket? It may seem as if the rocket lifts off the ground by pushing on the ground or the air behind it, but this is not the case. The rocket actually pushes on the gas and in turn the gas pushes on the rocket! If we add a hole at the bottom of the gas chambers, the gas can escape and thus exerts a reaction force forward on the rocket.
Now we have some idea of how our engine works, but how can we be sure of if and when the gas escapes? If it doesn't, we have little (read: no) chance of our rocket moving at all.
Finding the exact position and velocity of the H2 particles at all times may seem like a good way to know if and when they escape, but considering how there are millions of particles that move with seemingly random velocities, this would take up quite a bit if time. Unfortunately we can't afford supercomputers, and the two of us would like to launch our rocket before we die of old age.
Luckily, there are a few assumptions we can make in order to simplify how the gas moves, and we can attempt to make the situation less complicated by treating our gas as if it were what is referred to as an ideal gas. This gas works as an approximation that helps us model and predict the behavior of real gases by having a few characteristics that simplify how gas behaves:
First, we assume that all the gas particles behave as point particles, that is, they lack spatial extension. Imagine them as little dots, like the period at the end of this sentence. So long as the pressure doesn't get too big, the size of the particles will be much smaller than the average size between particles, and this is a close approximation to reality. Also, when the particles are this small and moving at such high speeds, we will do ourselves another favor and assume we can ignore the effects of gravity on their movement.
Particles that escape transfer momentum to the rocket.
Second, we ignore the fact that the particles interact with each other. The only interaction will be what we refer to as an elastic collision with the walls of the container. As a consequence of this the total momentum is conserved! As you may know, momentum can be described by velocity and mass:
\(\vec{P} = m\vec{v}\)
When the momentum is conserved, the total momentum in a closed box is zero. Adding an opening on the bottom allows us to use the change in momentum to transfer force to the rocket, as force can be described as change in momentum over time:
\(\vec{F} = \frac{d\vec{p}}{dt}\)
When we assume our gas behaves as an ideal gas, we are allowed to describe the pressure inside the engine by using something called the ideal gas law:
\(P = nkT\)
From this law we see how the pressure inside our engine can be described by the number of particles present, temperature, and a constant called the Boltzmann constant. Note that the higher the temperature, the higher the pressure, and when the temperature is high, the particles move faster which allows for more momentum to be transferred to the rocket. Because we keep the pressure and temperature constant, we know that as one particle leaves the engine, another has to take its place!
The ideal gas law didn't come from nowhere, you can derive it from the more general expression for pressure, \(P = \frac{1}{3}\int_0^{\infty}pvn(p)\: dp\), where \(n(p)\: dp\) is the number of particles per volume with a certain velocity. Knowing how to do this isn't necessary for understanding the rest of the blog, and so we will leave it on the board for the especially interested!
