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

1 2 x  x o  v o t  2 at

v  vo  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  v o t

v  v o 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.