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9.5 Energy in Orbital Motion 291

2
2
24
6.67 10 11 N m /kg 5.98 10 kg 1300 kg

2
1 1
¢ 7 6 ≤
4.23 10 m 6.65 10 m
10
3.29 10 J
This energy was supplied by the booster rocket of the satellite.

For an elliptical orbit, the total energy is also negative. It can be demonstrated that
the energy can still be written in the form of Eq. (9.24), but the quantity r must be
taken equal to the semimajor axis of the ellipse.The total energy of the orbit does not
depend on the shape of the ellipse, but only on its larger overall dimension. Figure
9.22 shows several orbits of different shapes but with exactly the same total energy. Total energy does not depend on
From Eq. (9.24) we see that if the energy is nearly zero, then the size of the orbit the shape of the ellipse, only on
the length of the semimajor axis.
is very large (note that E S 0 as r S ). Such orbits are characteristic of comets, many
of which have elliptical orbits that extend far beyond the edge of the Solar System FIGURE 9.22 Orbits of the same total
(see Fig. 9.23). If the energy is exactly zero, then the “ellipse” extends all the way to energy. All these orbits have the same semi-
infinity and never closes; such an “open ellipse” is actually a parabola (see Fig. 9.24). major axis.

Faye Halley
For a zero-energy
orbit, v 0 as
r .
orbit of
Biela Saturn
Encke
Winnecke
Orbits of negative total energy
near zero are large ellipses.

FIGURE 9.23 Orbits of some periodic comets. FIGURE 9.24 Orbit of zero energy—
a parabola.
Equation (9.21) indicates that if the energy is zero, the comet will reach infinite dis-
tance with zero velocity (if r , then v 0). By considering the reverse of this
motion, we see that a comet of zero energy, initially at very large distance from the
Sun, will fall along this type of parabolic orbit.
If the energy is positive, then the orbit again extends all the way to infinity and again
fails to close;such an open orbit is a hyperbola.The comet will then reach infinite distance
with some nonzero velocity and continue moving along a straight line (see Fig. 9.25).

A meteoroid (a chunk of rock) is initially at rest in inter-
EXAMPLE 10
planetary space at a large distance from the Sun. Under
For a positive-energy
the influence of gravity, the meteoroid begins to fall toward the Sun along a straight orbit, comet continues
radial line. With what speed does it strike the Sun? The radius of the Sun is with nonzero v as r .
8
6.96 10 m.
FIGURE 9.25 Orbit of positive energy—
SOLUTION: The energy of the meteoroid is a hyperbola.
GM m
S
1
2
E K U mv [constant] (9.25)
2 r