Basic rocket propulsion

Newton stated in his third law that every action must have an equal and opposite reaction. This has proved extremely useful for rocket propulsion as the law demonstrates that if a large amount of fuel is rapidly ejected from the back of a vessel it is quite clear that there is a large force acting on the fuel, which therefore, will also be acting on the vessel. This can be seen as conservation of momentum as force can be written as the change in momentum with respect to time. Therefore if there is always an equal and opposite reaction force, momentum is always conserved.
Imagine a rocket in a space without gravity at time t. It has a mass M and a velocity V. It turns on its thrusters for a short amount of time, delta t, causing an increase in velocity, and a decrease in its mass as shown.
rocket propulsion2
There is no change in total momentum of the system so we can write that:
Where P2 is the final system momentum and P1 is the initial system momentum. This can then be written as.
However as delta t tends towards zero, delta v divided by delta t tends towards dv/dt, the acceleration of the body. Also delta M divided by delta t tends towards -dM/dt, negative because delta M is negative. When rearranged this gives:
Where (u-(v+ delta v) is rewritten as the velocity of the gas flow relative to the rocket. At this point we can reach the basic equation for the thrust provided by a rocket propellant.
Where m(dv/dt) has become F as F=ma, relative velocity has been written as Ve, the exhaust gas ejection speed, and the change in mass with respect to time is q, the mass flow rate. The final term, (Pe-Pa)Ae, is required as there is an additional thrust force produced by the difference in pressure between the propellant and the gas of the atmosphere. Here Pe equals the pressure of the exhaust gas and Pa equals the ambient pressure. Therefore the pressure multiplied by the exhaust area Ae is the force provided by this term, the pressure thrust. This plus the momentum thrust is the total thrust provided by a rocket in the general rocket thrust equation.
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