EXPONENTIAL AND LOGARITHMIC FUNCTIONS

PROBLEM 8.

It is apparent from the rocket equation that the burnout velocity increases when the mass ratio increases. We can get a higher mass ratio by using a solid propellant because the stiff, rubberlike propellant mass serves as part of the structure. If no payload, or a very small payload, is included, a solid-propellant rocket could have a mass ratio of about 19. A typical average exhaust velocity for a solid propellant might be about 2.4 km per second. Could this launch vehicle achieve a 160 km Earth orbit?

Solution: Using the rocket equation,

v = 2.5 In 19 km/s

= (2.4) (2.94)

= 7.1 km/s,

which is much less than that needed for orbit.

The solution to the problem pointed out in the preceding examples is to use staging. That is, the launch vehicle is divided into two or more parts, or stages. As soon as the propellant has been burned in the first stage, there is a brief coast during which the heavy motors and structure in the first stage are jettisoned and permitted to fall into the ocean. Freed from this deadweight, the second-stage motors are much more effective; the same procedure is repeated for the remaining stages.

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