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Rocket equation

2010-04-09T21:04:28Z

Techer: /* Relativistic 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 ''ln''R

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

''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 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>\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.

==External Links==

*[http://en.wikipedia.org/wiki/Rocket_equation Wikipedia Article: Tsiolkovsky Rocket Equation]

Rocket equation

2009-10-22T23:23:38Z

Techer:

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

vf - vi = ve ''ln''R

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

''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 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.

==External Links==

*[http://en.wikipedia.org/wiki/Rocket_equation Wikipedia Article: Tsiolkovsky Rocket Equation]

Rocket equation

2009-10-18T21:44:40Z

Techer:

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

vf - vi = ve ''ln''R

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

''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 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'' ( ''ln'' R)

''tanh'' is the hyperbolic tangent function.

==External Links==

*[http://en.wikipedia.org/wiki/Rocket_equation Wikipedia Article: Tsiolkovsky Rocket Equation]