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

1

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

2

Tsiolkovski Formula

v 2 − v1 = v p .Log e

m1
m2

∆ v = v p .LN . mm12

3

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

4

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

5
6

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

7

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 


8

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

9

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

10
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