Rocket equation
The rocket equation is one solution to the momentum method of calculating a rocket's delta v. The equation is:
vf - vi = ve lnR
where:
<math>\Delta</math>v = vf - vi vf = Final Velocity of Rocket vi = Initial Velocity of Rocket ve = Exhaust Velocity of the rocket R = Mass Ratio of Rocket
lnR is the natural logarithm of R.
Mathematics
The rocket equation is derived from the conservation of momentum.
<math>\Sigma p_{i} = \Sigma p_{f} = Constant </math>
If mass is exhausted from a rocket engine at some mass flow rate m'e and some exhaust velocity ve, then that exhaust steam carries momentum, manifested as thrust:
<math>{dp \over dt} = m'_{e} v_{e} = v_{e} {dm \over dt} = m {dv \over dt}</math>
Using the algebraic representation of derivatives, this differential equation can be reduced to:
<math>dv = v_{e} {dm \over m}</math>
If the exhaust velocity is constant, then the integral of this is the rocket equation.
<math>\Delta v = \int dv = v_{e} \int {dm \over m} = v_{e} ln (R)</math>
Conservation of momentum can be applied to more propulsion situations than the simple case presented above. For example, under the influence of constant gravity:
<math>{dp \over dt} = m'_{e} v_{e} + g cos( \theta )</math>
Using the substitutions listed above, the integral of this relationship becomes:
<math>\Delta v = v_{e} ln (R) + g t cos( /theta )</math>
If the exhaust consists of two separate gas streams whose mass flow rates keep some constant ratio, r, to each other:
<math>r = {m'_{1} \over m'_{2}}</math>
<math>v_{avg} = { v_{1} m'_{1} + v_{2} m'_{2} \over m'_{1} + m'_{2}}</math>
Thus:
<math>v_{avg} = { r v_{1} + v_{2} \over r + 1 }</math>
and vavg can be substituted into the rocket equation rather than ve.
Example
Relativistic Rocket Equation
The rocket equation for speeds that become relativistic (~0.1c) is:
<math>\Delta</math>v \over c = tanh ( {ve \over c} ln R)
tanh is the hyperbolic tangent function.
