# Rocket equation

The rocket equation is one solution to the momentum method of calculating a rocket's delta v. The equation is:

v_{f}- v_{i}= v_{e}lnR

where:

<math>\Delta</math>v = v_{f}- v_{i}v_{f}= Final Velocity of Rocket v_{i}= Initial Velocity of Rocket v_{e}= Exhaust Velocity of the rocket R = Mass Ratio of Rocket

*ln*R 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 v_{e}, 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 v_{avg} can be substituted into the rocket equation rather than v_{e}.

## Example

## Relativistic Rocket Equation

The rocket equation for speeds that become relativistic (~0.1c) is:

<math>\frac{\Delta v}{c} = \tanh(\frac{v_e \ln(m_i/m_f)}{c\sqrt{1-\frac{v_e^2}{c^2}}})</math>

*tanh* is the hyperbolic tangent function.