EXPONENTIAL AND LOGARITHMIC FUNCTIONS

PROBLEM 9.

Let us design a two-stage vehicle to place a payload into Earth orbit. We shall make some simplifying assumptions to make this problem easier while preserving the basic idea: (l) the structure weight of each stage is 10 percent of the fuel weight, the remaining weight being payload; (2) the gain in velocity is divided equally among the stages, each contributing 4.5 km/s to the required final velocity of 9.0 km/s; (3) all stages use the same propellant with an exhaust velocity of 3.7 km/s. This third assumption is generally not true in practice‹for example, the Space Shuttle uses solid rocket boosters in addition to the main engines‹but our goal here is to see how staging works. For the sake of having a numerical example, we shall also assume that the total weight at liftoff is 5.0 x 10^4 kg. For this numerical example, determine the weight of fuel to be carried by each stage, the structural weight of each stage, and the weight of the orbital payload.

Solution: Let F1, S1, P1 represent fuel, structure, and payload weight, respectively, of the first stage, and F2, S2, and P2 those of the second stage. Since the "payload" of the first stage includes the entire second stage and the orbital payload,

P1 = F2 + S2 + P2

First Stage:

v = c ln R1

4.5 = 3.7 ln R1

ln R1 = 4.5/3.7 = 1.22

R1 = e^1.22 = 3.4

so

(F1 + S1 + P1)/(S1 + P1)

= (5.0 x 10^4)/(S1 + P1) = 3.4

Then

S1 + P1 = (5.0 x 10^4)/3.4 = 1.5 x 10^4 kg.

and

F1 = (5.0 - 1.5) x 10^4 = 3.5 x 10^4 kg

By assumption 1,

S1 = 0.10(3.5 x 10^4) = 3.5 x 10^3

Then

P1 = 1.5 x 10^4 - 3.5 x 10^3 = 1.15 x 10^4 kg.

Second stage: We again have, from the rocket equation,

4.5 = 3.7 ln R2

so

R2 = 3.4

Also,

R2 = (1.15 x 10^4)/(S2 + P2)

Then

S2 + P2 = (1.15 x 10^4)/3.4 = 3.4 x 10^3 kg

Therefore,

F2 = 1.15 x 10^4 - 3.4 x 10^3 = 8.1 x 10^3 kg

S2 = 0.10 (8.1 x 10^3) = 0.8 x 10^3 kg

P2 = (3.4 - 0.8) x 10^3 kg = 2.6 x 10^3 kg.

Our design for the two-stage launch vehicle may be checked as follows:

EXPONENTIAL GRAPHIC 9A.

Thus, although the single-stage launch vehicle discussed in Problem 7 could not place any payload into orbit, this two-stage vehicle can place nearly 5 percent of its weight into Earth orbit.

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