Projectile and Satellite Motion. Projectile Motion If there were no gravity a tossed object would...
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Projectile and Satellite Motion
Projectile MotionIf there were no gravity a tossed object would
follow a straight-line path and head out into space. However, because gravity gives an object weight, the object’s path curves toward the ground. The path forms the shape of a parabola.
The cannonball would follow the top path if there were no gravity.
Horizontal ProjectileA projectile is defined as an object that is launched by
some means, but continues on its own inertia. If a projectile is launched horizontally, it would continue as described in the diagram below. The top path would be its path without gravity.
Consider a cannonball projected horizontally by a cannon from the top of a very high cliff. In the absence of gravity, the cannonball would continue its horizontal motion at a constant velocity. This is consistent with the law of inertia. And furthermore, if merely dropped from rest in the presence of gravity, the cannonball would accelerate downward, gaining speed at a rate of 10 m/s every second. This is consistent with our conception of free-falling objects accelerating at a rate known as the acceleration of gravity.
If we project the cannonball horizontally in the presence of gravity, then the cannonball would maintain the same horizontal motion as before - a constant horizontal velocity. Furthermore, the force of gravity will act upon the cannonball to cause the same vertical motion as before - a downward acceleration. The cannonball falls the same amount of distance as it did when it was merely dropped from rest. However, the presence of gravity does not affect the horizontal motion of the projectile.
The projectile travels with a constant horizontal velocity and a downward vertical acceleration. The horizontal and vertical motions of the projectile are independent of each other. The object will fall just as if it were dropped straight down. For a horizontally launched projectile there is no acceleration in the horizontal direction. The object accelerates in the vertical direction only.
Upwardly Launch ProjectilesIf a cannonball is shot in an upward direction without gravity, the cannonball will follow a straight-line path and continue into space on its own inertia. The cannonball will really follow a parabolic path due to gravity. The cannonball will fall beneath the imaginary straight-line based on the equation:
d = ½ gt2
The “t” is the time the cannonball is in the air.
Without gravity the projectile will follow the straight line. With gravity it will fall below the line based on the equation: d = 5t2. The “t” is the time the projectile is in the air.
The d = 5t2 equation also applies to an object in free-fall. A vertical motion of an upwardly launched projectile behaves just as if it were tossed straight up. The same equations apply. The projectile slows down on the way up just as if it were tossed straight up, and get faster on the way down likewise.
The projectile still falls below its gravity-free path by a vertical distance of d = 5t2. However, the gravity-free path is no longer a horizontal line since the projectile is not launched horizontally.
The trajectory of an upwardly launched projectile looks like a parabola. The velocity slows on the way up. At point “x” the vertical velocity is “0”. The horizontal velocity is the same throughout its flight.
The earth’s curve is such that for every 8000 m or 8km the ground drops approximately 5 m. That means if you traveled in a perfectly straight line(not just walking forward)for 8 km the ground would be 5 m beneath you. If you throw an object horizontally it will follow a parabolic path toward the ground because of gravity.
Suppose a pitcher could throw a baseball that could travel 8000 m horizontally in one second. That means that gravity would cause the ball to fall 5 m in that one second. The baseball would be following the curve of the earth. The baseball will never hit the ground. The baseball will keep falling around the earth instead of into it. The baseball has become a satellite.
A projectile launched horizontally with a speed of 8000 m/s will be capable of orbiting the earth in a circular path; this assumes that it is launched above the surface of the earth and encounters negligible atmospheric air resistance. As the projectile travels horizontally a distance of 8000 meters in 1 second, it will drop approximately 5 meters towards the earth; yet, the projectile will remain the same distance above the earth due to the fact that the earth curves away at the same rate. If shot with a speed greater than 8000 m/s, it would orbit the earth in an elliptical path.
The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The velocity of the satellite would be directed tangent to the circle at every point along its path. The acceleration of the satellite would be directed towards the center of the circle - towards the central body which it is orbiting. And this acceleration is caused by a net force which is directed inwards(towards the center of the earth)in the same direction as the acceleration. Diagram on next slide.
Because of friction or drag in the atmosphere, a satellite cannot achieve orbit near the earth’s surface. This is approximately 150 km above the earth’s surface.
Occasionally satellites will orbit in paths which can be described as ellipses if they are shot faster than 8000 m/s. In such cases, the central body(the earth)is located at one of the focal points of the ellipse. Similar motion characteristics apply for satellites moving in elliptical paths. The velocity of the satellite is directed tangent to the ellipse. The acceleration of the satellite is directed towards the focus of the ellipse.
Tycho Brahe was a Danish astronomer given his own island by the Danish king in order to set up an observatory. Brahe spent 20 years measuring planetary positions. Brahe’s measurements were so accurate that his measurements are still used today. Upon Brahe’s death, his assistant Johannes Kepler continued Brahe’s work and made observations which became known as Kepler’s Laws of Planetary Motion.
Kepler’s first law of planetary motion is the observation that the planets’ orbit around the sun is not circular. The planets orbit the sun in an elliptical path with the sun as one of the focal points. The closer to the sun the planet is the more circular the orbit. Mercury’s orbit is nearly circular, but further planets like Mars and Earth have very elliptical orbits.
Kepler observed that the planets moved faster in their orbits when they were closer to the sun, and more slowly when they were farther away. During the winter months the earth is closer to the sun than during the summer. During the winter months the earth is moving faster, which is why winter is a shorter season than summer by a few days. Kepler described this in his second law:
An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The areas of the shaded areas are all equal.
In an elliptical orbit part of the path brings the satellite, or planet, closer in its orbit. As the satellite, or planet gets closer its speed increases, and consequently, its speed decreases as it moves farther away.
At point “A” the satellite is moving at its fastest speed. At point “B” the satellite is moving at its slowest speed.
Point “B” is called the apogee-the farthest part of the orbit. Point “A” is the perigee-the closest part of the orbit.
Once the satellite is in orbit it has two types of energy: kinetic and potential. The potential energy is based on its position in orbit. The kinetic energy is based on its velocity.
The “law of conservation of energy” demands that the total energy of the satellite remain the same throughout the orbital path. This means that, at any point in the satellite’s orbit, the kinetic energy + the potential energy will be the same. As the satellite moves through its orbits they both change. At points where the potential energy is increasing, the kinetic energy is decreasing and vice versa. However, sum of the two will always be the same. The kinetic energy is at its maximum where the speed is greatest, which is the closest point or the perigee. The potential energy is greatest at the farthest point, which is the apogee.
Kepler’s third law states: The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. This law applies to satellites and moons, not just to planets orbiting the sun. Both Galileo and Kepler were familiar with each other’s work, but neither accepted the other’s findings. Eventually, Isaac Newton would unite Galileo’s and Kepler’s work.
As seen before, if a projectile is thrown horizontally at 8000 m/s, above the effects of air drag, the projectile will achieve a circular orbit. If the speed is greater than 8000 m/s the orbit will be elliptical. What if the projectile is thrown vertically? At 8000 m/s the projectile would reach a certain height then come back down. There is a speed in which a projectile launched vertically could escape the gravitational field of the earth, this is called the escape speed or escape velocity.
Escape speed depends on factors similar to those of gravity. Primarily, escape speeds depend on the mass of the planet from where it is launched, but also the distance from the center of the planet from which it is launched. Distance doesn’t become a major factor unless the launch pad is very high like at the top of very tall mountain. The most important factor affecting escape speed is the mass of the planet from which it is launched. Escape speed from the earth is 11.2 km/s(approx. 25000 mph).
Once the engines of the rocket accelerate it to 11.2 km/s, the engines shut off, and the rocket will leave the atmosphere on its own momentum. The rocket will also leave the atmosphere at lower speeds, but only if the engines continue to do work. The problem with this is that it requires more fuel and would be more costly.
The escape speed for an projectile launched from the moon is 2.4 km/s. While the escape speed from Jupiter is 60.2 km/s. Even if a rocket escapes the atmosphere of the earth would the rocket continue on and eventually leave the solar system? The rocket would have to the escape the gravitational field of the sun. The speed needed to escape the sun’s gravitational field from the distance of the earth’s orbit is 42.2 km/s. If the rocket does not reach 42.2 km/s, it will leave the earth’s atmosphere, but it will remain in the solar system and continue to orbit the sun in an elliptical path.
Pioneer 10 was the first probe to escape the solar system. Pioneer 10 was launched in 1972 with a launch speed of 15 km/s. This is enough to escape the earth’s gravitational field but no the sun’s. To achieve the 42.2 km/s to escape the solar system, the probe was sent toward Jupiter. The probe’s path took it around Jupiter using the planet’s gravitational field like a slingshot. The probe headed toward edge of the solar system at a speed of over 42 km/s. Pioneer 10 passed the orbit of Pluto in 1984, and will continue to drift in space unless it is hit by something.