Problem 1 (10 points)
Consider the design and optimization of a 2 stage rocket. We desire to determine the optimum mass
ratio between the first and second stages of a rocket. The payload is 10% of the total mass of the
rocket M. The mass of the first and second stages together is then 90% of the total mass M. 90% of
the mass of each stage is fuel. The mass of the first stage is R times the combined mass of the two
stages, (m1 = .9 ∗ R ∗ M), leaving (1 − R) times the combined mass of the two stages for the mass of
the second stage (m2 = .9 ∗ (1 − R)M). (R is a number less than 1, say .55 as a typical value.) Now
let’s launch. Take c = 4500 and for convenience take the total mass M equal to 1 (payload plus both
stages; M = 1).
Ignore gravity and use the expression for the ”ideal” Δv from a rocket firing from Lecture 14,
Δ(v)ideal = −cln( mf ) = −cln(µ). Calculate the velocity after the first stage is fired to comple-
m0
tion and about to be thrown away. Then calculate the additional velocity from the firing of the upper
stage after the first stage has been discarded.( Using (v0)ideal ignores the effect of gravity and gives
the most optimistic prediction of final velocity.)
1-a) Now fire the first stage and fly it to completion. For a general value of R, and the specified
masses of the payload, fuel and second stage, what is the final velocity of the rocket after the
first firing? What is µ for this part of the calculation?
1-b Now fire the second stage, and fly it to completion. What is the final velocity of the rocket
vf after this firing. Express this final velocity as a function of R, the mass ratio of the first
stage m1 to the total rocket mass m1 + m2 (excluding payload). What is µ for this part of the
calculation?
1-c Find the optimum value of R that maximizes the final velocity. You may do this by either
plotting vfinal (R) vs. R, or by taking ( dv )final, setting it equal to zero, and solving for R.
dR
1-d Compare this ”optimum” velocity to the final velocity for a single stage rocket. (What would
R be for a single stage rocket?)
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