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Projectile and Satellite Motion
The term “projectile motion” here means two-dimensional motion, motion in a
plane. The principles can easily be extended to three-dimensions, however. The
situation we want to investigate most closely is the free fall example, sometimes
called “ballistic” motion. The assumption are the same as before – near the
surface of the Earth (so that the acceleration of gravity is constant) in the
absence of air resistance (so that gravity is the only force). Rather than treat the
motion itself, we will break the motion into vertical and horizontal components.
The problem is then one of two linear motion problems linked by time. The
equations
x  x o  v o t  12 at 2
v  v o  at
still apply, since the acceleration is constant. In fact, the strong advantage of
treating the motion in this way is the gravity only acts vertically. There is no
horizontal force and, therefore by Newton’s 2nd Law, no horizontal
acceleration. Our two equations then reduce to
x  x o  vo t
v  vo
The velocity on the horizontal direction never changes.
In the vertical direction we have the acceleration due to gravity. We can treat
the horizontal and vertical motions as independent. So if two balls are launched
together, one horizontally and the other simply dropped, they will hit the ground
at the same time, because they both start with zero vertical velocity and
accelerate at the same rate.
After treating the horizontal and vertical components of the motion, we can
then combine the two to produce the true path of the object. The path turns out
to be approximately a parabola. We need to say approximately here, because
we are dealing with certain assumption in free fall that will not be met in the real
situation. The parabola is one of a class of curves called “conic sections,”
because they are all derived by cutting a right circular cone. The shapes are the
circle, ellipse, parabola, and hyperbola. We will return to these shapes again
shortly.
One of the interesting consequences of projectile motion is that two different
launch angles can be used to achieve the same range (horizontal distance at
which the object lands measured from the starting point). The sum of these two
launch angles is always 90°. Thus two launches at 30° and 60° will put the
projectile at the same point when it lands. The object launched at 60° rises
higher and takes longer to land, but the range is the same as the projectile at
30°. The launch angle that produces the greatest range is 45°.
The other conclusions we reached in our previous discussion of free fall still
hold, i.e., the rise time equals the fall time, the speed at the end is the same as
the speed at the beginning.
If we relax the situation of free fall and consider the situation with air
resistance, the path becomes more complex. We can, however, say that the
projectile has a smaller range, because air resistance always opposes motion.
The situation of satellite motion is simply high speed projectiles. The satellite
is constantly falling without getting closer to the ground. Since satellite
motion is very similar to planetary motion, let’s consider the general problem.
Planetary motion had been considered since the time of the ancient Greeks. In
those days people believed that the paths of the planets were perfect circles
going around the Earth. The work of five individuals during the late Renaissance
changed our thinking about the paths of the planets and the structure of the solar
system. These were: Nicholas Copernicus, Tycho Brahe, Johannes Kepler,
Galileo Galilei, and Isaac Newton. We have already discussed Galileo and
Newton.
Copernicus taught Aristotelian philosophy at the University. He thinks the
old model of Ptolemy is too complex to be correct. Copernicus believes the
heliocentric (Sun-centered) model must be correct. The Earth is now placed as
the third planet moving around the Sun.
Tycho Brahe was a late 16th century Danish nobleman who carried out an
extensive observing program of the planets. He believed that only through
observations could we discern one model from another. After being expelled
from Denmark and settling in Prague, he hired Johannes Kepler to show what
the orbits of the planets were.
Johannes Kepler devised the very first natural laws with his laws of
planetary motion. The first law showed that planets orbit the Sun in elliptical
paths, the Sun being at one focus of the orbit. The second law tells us how
the planets move on their orbits - faster closer to the Sun, slower farther
away. The third law relates the orbital period to the size of the orbit. The
laws of planetary motion were empirical and universal, although Kepler never
correctly surmised the cause of the orbits.
The universality of the Kepler’s Laws allows us to use them to discuss
satellite motion. The first law then tells us that satellites orbit the Earth in
ellipses, with the center of the Earth at one focus of the ellipse. If we imagine a
horizontal projectile fired at increasing initial speeds, we would see that the path
changes from a highly eccentric ellipse to a perfect circle as the launch speed
approached orbital velocity (7 km/sec). For this thought experiment we assume
that the Earth’s surface does not impede the motion. As the launch speed
increases beyond orbital velocity, the path once again becomes elliptical. When
the launch speed reaches escape velocity (11 km/sec), the path becomes a
parabola. For launch speeds higher than escape velocity, the path is a
hyperbola. All of the classic conic sections are represented as possible orbital
shapes under the influence of gravity.