Over 100 years ago, Konstantin Tsiolkovsky, father of modern rocketry, derived an equation that would help lift humanity into the heavens. This is his famous rocket equation.
Newtons second law is F=ma. This can also be written intFdt=ma=dp/dt. The final speed of the rocket will be determined by the initial mass of the rocket and fuel, final mass of the rocket, the exhaust velocity, and the gravitational losses.
We will write Newtons second law as -mg=mdv/dt+vedm/dt where m is the instantaneous mass of the rocket, ve is the exaust velocity, dv/dt is the acceleration. and dm/dt is the rate of consumption of fuel. We can now divide by m on each side and multiply by dt on each side. This will result in -gdt=dv-(ve/m)dm. Now set the bounds and integrate dt from 0 to tf, dv from v0 to tf, and dm from m0 to df. This will result in -gtf=deltav +veln(mf/m0) which we can re-arrange to deltav=-veln(mf/m0)-gtf.
Lets examine this. deltav will be the change in velocity, which should be positive. Since mf/m0 is a fraction less than 1, the ln of it will be negative. Multiplying a negative by -ve will yield a positive, which is what we want. Now we take this number and subtract the velocity losses from gravity, which is gtf. This is known as the ideal rocket equation which states that the change in velocity depends on the burn time, exhaust velocity, and mass ratio of an empty to fully fueled rocket. It is called ideal because it assumes no drag and that the rocket doesn’t change direction.
oh no rocket science! 