FIRST STEPS TOWARD SPACE - Smithsonian Institution Libraries

NUMBER 10
In the present case, there are no difficulties in theory. To alter the vehicle's direction,
one need only incline the propulsor in such a way that the direction of the force
it develops would be at an angle with the trajectory. If the displacement of the
propulsor was not sufficient to obtain rotation in all directions, one or two smaller
auxiliary propulsors would be enough to obtain complete maneuverability.
II
To remove a heavy body from the attraction of a planet, one has to spend energy.
Let us consider a mass M at a distance x from the center of a planet whose radius
is R. Let 7 be the acceleration of gravity at the surface of this planet. To move the
body away a distance dx, it will be necessary to do an element of work
which gives
dZ = — My — dx
Z = MyR (-3
We can readily see that to move a given mass to infinity the necessary work to be
done would be finite and given by
x 2
Z = MyR
Or if we let P be the weight of the body at the surface of the planet, then
Z = PR
We also see that if we consider the weight of the body as the result of the principle
of universal attraction applied to body and planet, we can write after letting U
denote the planet's mass
MU
This gives for expressing the work necessary for removal of the body to infinity
MU
Z = k
R
Therefore, if we give initially to a body on the surface of a planet a sufficient velocity
to remove from the planet, this body would increase its distance indefinitely.
For the earth, the minimum velocity would be 11,280 m/s, i.e., a projectile launched
from the earth with a velocity larger than 11,280 m/s (not considering air resistance)
would never fall back.
This critical velocity is exactly the same as that which a body would acquire falling
toward the earth from infinity and having no initial velocity with respect to the planet.
The motion of such a body would be given by the equation
R 2
V 2 = 2g —
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