192
Self-Check
Which of Kepler’s laws would it make sense to apply to hyperbolic orbits.
Discussion Questions
A. How could Newton find the speed of the moon to plug in to a=v
2
/r.
B. Two projectiles of different mass shot out of guns on the surface of the earth
at the same speed and angle will follow the same trajectories, assuming that
air friction is negligible. (You can verify this by throwing two objects together
from your hand and seeing if they separate or stay side by side.) What
corresponding fact would be true for satellites of the earth having different
masses.
C. What is wrong with the following statement. "A comet in an elliptical orbit
speeds up as it approaches the sun, because the sun’s force on it is increas-
ing."
D. Why would it not make sense to expect the earth’s gravitational force on a
bowling ball to be inversely proportional to the square of the distance between
their surfaces rather than their centers.
E. Does the earth accelerate as a result of the moon’s gravitational force on it.
Suppose two planets were bound to each other gravitationally the way the
earth and moon are, but the two planets had equal masses. What would their
motion be like.
F. Spacecraft normally operate by firing their engines only for a few minutes at
a time, and an interplanetary probe will spend months or years on its way to its
destination without thrust. Suppose a spacecraft is in a circular orbit around
Mars, and it then briefly fires its engines in reverse, causing a sudden de-
crease in speed. What will this do to its orbit. What about a forward thrust.
10.3Apparent Weightlessness
If you ask somebody at the bus stop why astronauts are weightless,
you’ll probably get one of the following two incorrect answers:
(1) They’re weightless because they’re so far from the earth.
(2) They’re weightless because they’re moving so fast.
The first answer is wrong, because the vast majority of astronauts never get
more than a thousand miles from the earth’s surface. The reduction in
gravity caused by their altitude is significant, but not 100%. The second
answer is wrong because Newton’s law of gravity only depends on distance,
not speed.
The correct answer is that astronauts in orbit around the earth are not
really weightless at all. Their weightlessness is only apparent. If there was no
gravitational force on the spaceship, it would obey Newton’s first law and
move off on a straight line, rather than orbiting the earth. Likewise, the
astronauts inside the spaceship are in orbit just like the spaceship itself, with
the earth’s gravitational force continually twisting their velocity vectors
around. The reason they appear to be weightless is that they are in the same
orbit as the spaceship, so although the earth’s gravity curves their trajectory
down toward the deck, the deck drops out from under them at the same
rate.
The equal-area law makes equally good sense in the case of a hyperbolic orbit (and observations verify it). The
elliptical orbit law had to be generalized by Newton to include hyperbolas. The law of periods doesn’t make sense
in the case of a hyperbolic orbit, because a hyperbola never closes back on itself, so the motion never repeats.
Chapter 10Gravity