# Difference between revisions of "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:

$\Delta$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.

    $\Sigma p_{i} = \Sigma p_{f} = Constant$


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:

    ${dp \over dt} = m'_{e} v_{e} = v_{e} {dm \over dt} = m {dv \over dt}$


Using the algebraic representation of derivatives, this differential equation can be reduced to:

    $dv = v_{e} {dm \over m}$


If the exhaust velocity is constant, then the integral of this is the rocket equation.

    $\Delta v = \int dv = v_{e} \int {dm \over m} = v_{e} ln (R)$


Conservation of momentum can be applied to more propulsion situations than the simple case presented above. For example, under the influence of constant gravity:

    ${dp \over dt} = m'_{e} v_{e} + g cos( \theta )$


Using the substitutions listed above, the integral of this relationship becomes:

    $\Delta v = v_{e} ln (R) + g t cos( /theta )$


If the exhaust consists of two separate gas streams whose mass flow rates keep some constant ratio, r, to each other:

    $r = {m'_{1} \over m'_{2}}$

    $v_{avg} = { v_{1} m'_{1} + v_{2} m'_{2} \over m'_{1} + m'_{2}}$


Thus:

    $v_{avg} = { r v_{1} + v_{2} \over r + 1 }$


and vavg can be substituted into the rocket equation rather than ve.

## Relativistic Rocket Equation

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

    $\Delta$v \over c = tanh ( {ve \over c} ln R)


tanh is the hyperbolic tangent function.