This lesson will describe the concepts of circular velocity, escape velocity, open orbits, closed orbits, and geosynchronous satellites. It will provide the equations necessary for you to be able to calculate out circular and escape velocity as well.
Have you ever wondered how fast a satellite up in space has to be traveling to stay in circular orbit around the Earth?
The velocity an object needs to have in order to orbit another larger object in a circle is called circular velocity. An object traveling at circular velocity obviously travels in a circle, but one traveling just under or just over this velocity travels in an elliptical orbit. While our solar system bodies (namely the planets and their natural satellites) travel in very slightly elliptical orbits, we'll just say they're circular for simplicity's sake for this lesson.
Circular Velocity Formula
That being said, there's actually quite a simple formula for circular velocity on screen where it is equal to the square root of G, the gravitational constant of 6.67 X 10 ^-11 m^3/s^2kg, multiplied by M, the mass of the larger (central) object in kilograms, and divided by the radius, R, of the orbit in meters.
I used the word 'satellite' before, but that word doesn't necessarily refer to a man-made satellite. The Moon is Earth's natural satellite as well. Let's figure out the Moon's velocity then.
Earth has a mass of 5.98 X 10^24 kg, and the moon is in orbit around Earth 3.84 X 10^8 m from Earth's center. Just plug away to get an answer of 1,020 m/s, or about 2,278 mph.
Note how the equation shows you that circular velocity is inversely proportional to the square root of the radius. This means a smaller orbit will result in a faster speed because the force of gravity on the smaller object will be stronger.
To demonstrate this, you can tie a rope to a marble and spin it in a circle. The marble is like a small object in orbit around your hand, the bigger object. Start twirling the marble in front of you in a circle. Then, stick your finger out so that the string begins to loop around your finger. As it loops around your finger, the string gets shorter (R gets smaller), and the marble spins faster and faster until it hits your finger and stops.
While the Moon is a satellite of Earth, it isn't a geosynchronous satellite of Earth. A geosynchronous satellite is simply a satellite that remains above the same spot on Earth's surface. It does so by orbiting around the Earth in an eastward direction, completing its orbit every 24 hours. This happens at about 42,000 kilometers from the Earth's center. If the satellite was much closer to Earth's center, right above Earth's atmosphere, it could have an orbital period of only 90 minutes as it travels at 7,790 m/s, way faster than the moon.
Escape Velocity, Open & Closed Orbits
Circular orbits, and technically the slightly elliptical orbits of our solar system's planets and satellites, are all a type of closed orbit. This is a circular or elliptical orbit that returns to the same place in its orbit over and over.
If you spin that marble in a circle above your head and keep your hand steady, the marble will return to the same point in space over and over again as it circles around. It does so because it's in a closed orbit. It has nowhere to go but stay in that circle, which is, by definition, a closed shape.
In a circular orbit, the speed of an object is the same throughout its orbit. In elliptical orbits, the smaller object's speed is faster as it nears the larger object.
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But if an object travels as fast as or faster than escape velocity, then it will enter into an open orbit. Escape velocity is the smallest velocity a small body requires in order to escape from the gravitational attraction of a more massive body, and an open orbit is a kind of orbit where an object is carried away, never to return to its starting point. An object traveling at escape velocity will have a parabolic orbit, and one that travels faster than escape velocity will have a hyperbolic one. Both types of orbits are open orbits.
Just like circular velocity has an easy equation, so too does escape velocity. Here, escape velocity is equal to the square root of 2 X G X M all over R. Earth's mass from before is 5.98 X 10^24 kg, and its radius is 6.38 X 10^6 m. By plugging those numbers in, you get an escape velocity of 11.2 km/s. This is the escape velocity from the surface of the Earth.
Escape velocity isn't the same as circular velocity, so don't get confused.
The velocity an object needs to have in order to orbit another larger object in a circle is called circular velocity, while escape velocity is the smallest velocity a small body requires in order to escape from the gravitational attraction of a more massive body.
An object traveling at circular velocity has a circular orbit, but just above or just below this velocity, it will travel in an ellipse. A circle and an ellipse are a kind of closed orbit, a circular or elliptical orbit that returns to the same place in its orbit over and over.
Some objects, namely man-made satellites, are geosynchronous satellites in closed orbit around Earth. A geosynchronous satellite is a satellite that remains above the same spot on Earth's surface.
But if an object is traveling at escape velocity, it will move into a parabolic orbit. If it travels faster than escape velocity, it will move in a hyperbolic orbit. Both a parabola and a hyperbola are a type of open orbit, a kind of orbit where an object is carried away, never to return to its starting point.
After you have finished with this lesson, you'll be able to:
Differentiate between circular velocity and escape velocity
Describe a closed orbit
Define geosynchronous satellite
Explain the open orbits created by escape velocity
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