EXPONENTIAL AND LOGARITHMIC FUNCTIONS

PROBLEM 10.

Show that when all stages use the same propellant, the total mass ratio of a multiple-stage launch vehicle is equal to the product of the individual mass ratios.

Solution: Indicate the burnout velocities and mass ratios of the first, second, third stages, and so on, by the subscripts 1, 2, 3, and so on. Then, using a three-stage vehicle as an example,

v1+ v2 + v3 = c ln R1 + c ln R2 + c ln R3

v = c (ln R1 + ln R2 + ln R3)

v = c log[base e](Rl x R2 x R3)

(Note: Making the structure stronger so that it can support large payloads reduces the mass ratios. However, if we have several stages, the total mass ratio can become very high, producing much greater performance.)

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