EXPONENTIAL AND LOGARITHMIC FUNCTIONS

PROBLEM 7.

The theory of rocket flight shows that the velocity gained by a launch vehicle when its propellant is burned to depletion is expressed by the equation

v = c ln R

where:

v is the velocity gained by the vehicle during launch;
c is the exhaust velocity of the engine;
ln R is log[base e]R, or the natural logarithm of R;
and R is the mass ratio of the spacecraft defined by R = takeoff weight/burnout weight.

a. The takeoff weight consists of propellant or fuel, F, structure, S, and payload, P. At burnout, assuming all the fuel has been used, the remaining weight is S + P, so that R = (F + S + P)/(S + P). In general, the weight of fuel cannot be more than about10 times the weight of the structure in order for the vehicle to withstand the stresses of operation. Show that if F = lO S, then an upper limit for R is 11.

Solution: If F = lO S, then R = (F + S + P)/(S + P) = (10S + S + P)/(S + P)

= [ 11( S + P) - 10P]/(S + P)

= 11 - 10P/(S + P) less than or equal to 11.

So the largest possible value for R is 11, but we see that in order to actually achieve this value, it is necessary for P to be O‹in other words, the launch vehicle could carry no payload!

b. The minimum altitude for a stable orbit about Earth is about 160 km. At lower altitudes, air resistance slows the spacecraft and causes a rapid deterioration of the orbit. As will be shown in Problem 1 of Chapter 9, the spacecraft must attain a velocity of about 7.8 km per second to orbit at 160 km. However, in order to overcome the retarding effect of Earth's atmosphere while the spacecraft is ascending, the total velocity imparted by the launch vehicle must be at least 9.0 km/s. What is the minimum exhaust velocity needed by the rocket engine if R = 11?

Solution: Substituting

v = 9.0 km/s

and R = 11 in the rocket equation,

9.0 km/s = c ln 11

c = (9.0/ln 11)km/s = 9.0/2.4 = 3.8 km/s

c. The propellants used for engines such as those of the Delta, Centaur, and Saturn launch vehicles could produce exhaust velocities averaging at most 3 km/s, which would not be sufficient to achieve orbit. The main engines of the Space Shuttle use a mixture of liquid hydrogen and liquid oxygen, which can produce exhaust velocities of 4.6 km/s. However, in order for the Shuttle to perform its tasks and return to Earth with its crew, it has an R-value of around 3.5. Could the Space Shuttle achieve orbit with its main engines?

Solution: If

c = 4.6 km/s, and

R = 3.5, then

v =4.6 ln 3.5 km/s

= (4.6) (1.25) km/s

= 5.8 km/s,

which is not sufficient for orbit.

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