satellites and space vehicles moving
under the influence of forces such as
gravity, atmospheric drag, thrust, and so
on. It is a modern spin-off of celestial
mechanics, or the study of the motions of
planetary and celestial bodies. One of the
main scientists who built the foundations
of orbital mechanics was mathematician
Isaac Newton (1642–1727), who put forth
his laws of motion and formulated the
law of universal gravitation. (For more
about Newton, see “History of Mathemat-
ics” and “Mathematical Analysis”; for
more about Newton’s laws, see “Mathe-
matics in the Physical Sciences.”) Today’s
aerospace engineers apply orbital
mechanics to such problems as rocket
and spacecraft trajectories, reentry and
landing of space vehicles, rendezvous
computations (such as the Space Shuttle
to the International Space Station), and
lunar and interplanetary trajectories for
manned and unmanned vehicles.
How do engineers determine the escape velocityof a rocket?
A ball thrown into the air will rise and then return, thanks to the Earth’s gravity. If the
ball is given a larger initial velocity, it will rise even higher and then return. With even
more velocity, the ball will reach a certain escape velocity, in which the ball “escapes”
the gravitational pull of the planet. If the ball is launched with an initial velocity
greater than the escape velocity, it will rise and not return. In this case, physicists say
that the ball was given enough kinetic energy to overcome all of the negative gravita-
tional potential energy—or, it launches into space. Thus, if mis the mass of the ball, M
is the mass of the Earth, Gis the gravitational constant, vis the velocity, and Ris the
radius of the Earth, then the potential energy is equal to GmM/R. The kinetic energy of
the launched ball is equal to mv^2 /2. That means the escape velocity is equal to:
This is independent of the mass of the ball. To see how this works to an aerospace
engineer, just replace the word “ball” with “space vehicle.”
R2 GM345
MATH IN ENGINEERING
Without knowing the mathematics involved in
orbital mechanics, the international space station
would plunge back into the Earth instead of circling
it in a stable orbit. Stone/Getty Images.