Application of Tsiolkovski Formula to multi-stage rocket launcher systems.

Comparison of multi stage with single stage. Lets take a rocket at launch and analyse the proportion of mass taken by each of its principle parts. At launch whether the rocket is a single or multiple stage rocket the mass can be represented by

MLaunch = mstruct + mfuel + msat

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Now let the fuel used during the first stage burn be α then at the end of the first stage burn the rocket mass will be represented by M1

M1

= mstruct + (1-α)mfuel + msat

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Tsiolkovski Formula

v 2 − v1 = v p .Log e

m1 m2

∆ v = v p .LN . mm12

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Applying Tsiolkovski’s formula to the first stage we get a mass ratio of : m struct + m fuel + m sat  ML  =    M 1  First m struct + (1 − α ) m fuel + m sat

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Where α is the proportion of fuel mass used in the first stage and the structure mass is assumed to be proportional to fuel mass required.

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For the second stage we assume that the structure mass of the first stage has been removed following seperation

M2

= (1-α) mstruct + (1-α)mfuel + msat

M3

= (1-α) mstruct + msat

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So applying Tsiolkovski’s formula to the second stage we get a mass ratio of :

(1 − α ) mstruct + (1 − α ) m fuel + msat  M2  =   (1 − α ) m struct + msat  M 3  Second

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Tsiolkovski Formula

v 2 − v1 = v p .Log e

m1 m2

Now introducing the mass ratios of the two stages in the Tsiolkovski formula we have :

   M2  M   ∆ v1+ 2 = v p . LN . L  + LN .   M 1  First  M 3  Second  

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For a single stage M  ∆ v = v p. LN  0   M1 

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So the multiplying factor for using a two stage rocket launching system is given by the following   M   ML   + LN  2    LN  ∆ v1+ 2  M1   M3   = v p .   ∆v M    LN  0  M    1 Excel table with worked example

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