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The rocket equation is the governing equation for all of rocketry, it is derived from the conservation of momentum in a time varying mass system. That means that as the rocket uses more fuel, it gets lighter and therefore can accelerate faster. The rocket equation in its basic form is a differential equation, this equation can be manipulated and changed to include all possible forces; like gravity, drag, and many others. By including all of these forces one needs a computer to solve the problem, for our uses including no external forces, or just gravity is a very good approximation. The formula below is the rocket equation with no external forces.

Δv=b ln(m0/m)

Here Δv is the change in velocity, b is the gas velocity, m0 is the initial mass, and m is the mass without fuel. The gas velocity is related to the thrust, which is the basic measurement of a propellant. This says that the smaller m gets the bigger the change in velocity is, and therefore the faster the rocket moves. What matters here is the gas velocity and the size of m0/m. This can be rearranged as follows:

1-m/m0=1-exp(-Δv/b)

This is a more useful equation because we can find a percentage of the rockets mass that must be fuel for the rocket to reach a desired velocity. An example: To make it to low earth orbit a rocket must reach 9.7km/s Δv, so starting from rest and assuming a gas velocity of 4.5km/s, we get 1-m/m0=0.884, this says that 88.4% of the ships mass must be fuel, because 88.4% of the initial mass of the rocket was used up to achieve 9.7km/s and low earth orbit. This leaves 11.6% for payload and the rest of the ship.

To improve efficiency and the amount of payload a rocket can bring into orbit rockets use stages, this means that instead of one big fuel tank and 1 engine there are separate fuel tanks with their own engines and when the lowest tank is empty it detaches from the rocket. This is a typical setup, and a perfect example is the Saturn V rocket, which had 3 stages and an engine onboard the actual spacecraft. The rocket equation holds exactly the same as for a single stage but when the old stage is jettisoned the initial mass is reset to the total mass of the rocket minus the jettisoned stage. Here is an example of a 2 stage rocket with the same values as the single stage rocket. Suppose that the first stage should provide a Δv of 5.0 km/s; 1-Δv=0.671, therefore 67.1% of the initial total mass has to be propellant. The remaining mass is 32.9 %. After deposing of the first stage, a mass remains equal to this 32.9%, minus the mass of the tank and engines of the first stage. Assume that this is 8% of the initial total mass, then 24.9% remains. The second stage should provide a Δv of 4.7 km/s; 1-Δv=0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2%, and 8.7% remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7% is available for the payload and the rest of the ship. As you can see using two stages allows us to bring 5.1% more mass into space.