Propulsion efficiency of an engine
A rocket engine changes the velocity of a spacecraft by applying an impulse for a certain time :
F.dT = m.dV (1)
would be the basic equation. The impulse applied results in a change in momentum of the spacecraft. The function of the rocket engine would then be to apply a force F during a time t to change the velocity of a mass m in a quantity dV (delta V), which is determined by trajectory requirements (for a Mars insertion burn, in this case.
As the propellant is consumed, the mass of the rocket changes, and so the change in mass has to be taken into account.
If we consider the problem from the point of view of the propellant mass, the same equation applies to the mass dM of propellant
F.dt = dM.ve (2)
where ve is the exhaust velocity, the velocity with which the propellant leaves the rocket engine.
Finding F :
F = dM/dT*ve
Replacing F in (1) :
dM/dT*ve*dT = M*dV
Simplifying and separating variables :
dM/M = dV/ve
MV = (M)(V+dV)-dM.ve
MV = MV + dV M - dM ve
dM/M = dV/ve
Integral dM/M = Integral dV/ve
ln(M/Mo) = (Vf-Vo)/ve
M/Mo = e^(dV/ve)(3)
Specific impulse is defined as a practical way of relating the thrust of a rocket engine to the weight of propellant flow. This variable is more often used than the exhaust velocity for it relates the thrust of an engine to the mass of propellant and can be directly compared between different propulsion alternatives :
Isp = F/(dM/dt*g)
Where Isp is specific impulse (thrust per unit flow rate of propellant) and is measured in seconds.
Isp*g = F/(Dm/dt)(4)
In equation (2) :
F.dt = dM.ve
F/(dM/dt) = ve
Substituting (4) :
Isp. g = ve
(all this assumes g to be a constant, which is not always the case as we leave Earth orbit)
Replacing in (4) :
M/Mo = e^(dV/ve)
M/Mo = e^(dV/Isp*g)
which constitutes the rocket equation.
The rocket equation is extremely useful in choosing the propulsion system, for it will return the initial to final mass ratio (M/Mo) knowing the Isp of the engine to be utilized.
The more powerful rocket engines will have a higher thrust to weight ratio and thus a greater Isp. This results in a lower M/Mo, that is, that the ratio of final mass (after the propellant is burned) to initial mass (when the rocket is fully loaded with propellants) is lower, allowing for a lighter rocket.