The Navstar Global Positioning System (GPS) utilizes a group o...

Question

The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed around the Earth, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical mile = 1.852km = 6076ft].
(a) Determine the speed of each satellite.

(a) Determine the speed of each satellite.

Accepted Answer

Welcome back. Everyone in this problem. Scientists have spotted a large space debris orbiting the earth at an altitude of 13,000 nautical miles above the surface. Given that one nautical mile is 1.852 kilometers. The mass of the earth is 5.97 multiplied by 10 to the 24th kilograms. The radius of the earth is 6370 kilometers and the gravitational constant G is 6.67 multiplied by 10 to the negative 11th, Newton square meters per square kilogram. Find the speed of the space debris for our answer choices. A says that is 1.31 multiplied by 10 to the 6 m per second. B 3.62 multiplied by 10 to the third meters per second. C 2.07 multiplied by 10 to the negative 1 m per second. And the D 4.29 multiplied by 10 to the negative 2 m per second. Now, if we're going to find the speed of the space debris, let's first make a note of all the information that we have here. So so far, we're given the mass of the earth to be equal to 5.97 multiplied by 10 to the 24th kilograms. Ok. And the radius of the earth, we're also told that the radius of the earth, let's call that re is 6370 kilometers or in SI units, 6.37 multiplied by 10 to the sixth meters. Ok. We are already given the gravitational constant in SI units. So I'm gonna leave it here and we're told that the altitude of our space debris orbiting the earth is 13,000 nautical miles. So since one nautical mile is 1.852 kilometers, then 13,000 nautical miles is going to be 13,000 multiplied by 100 sorry 1852 m per nautical mile, which is going to give us 2.408 multiplied by 10 to the seventh meters. Ok? And now we have all of this information and we are supposed to use it to figure out the speed of our space debris. So the first question we have to ask ourselves is, what do we know about speed that's related to any of the information here. We'll recall that we can find the speed of an orbiting object V as being equal to the square root of GM divided by R where G is the gravitational constant M is the mass of the object, the sorry, the mass, the object is orbiting around and R is the distance between the object and our mass. In this case, we have our distance or radius of the earth and we have the height in nautical miles of our object. So if we think about it, let's say this is the earth, this is our space debris. Really. The distance that we want is from the center of earth, from the earth's center of mass all the way to the object which would be equal to the radius of the earth. OK. Plus plus our distance from the surface of the earth which is equal to each. So in this case, R is going to be equal to re divide. Sorry R plus H. So now we can go ahead and solve for V. OK? Because no, this means then that V is going to be equal to the square root. OK. The square root of our gravitational constant G which is 6.67 multiplied by 10 to the negative 11th. It's a lot of units here. Newton square meters per square kilogram. OK? Multiplied by the mass of the earth which is 5.97 multiplied by 10 to the 24th kilograms. OK? All divided by the radius of the earth which we said was 6.37 multiplied by 10 to the 6 m plus or height of our debris in meters. 2.408 multiplied by 10 to the seventh meters. And not to solve for V, we can put this expression into our calculator. When we do, we should get V to be equal to 3616.223 m per second. If we write it to three significant figures, like the rest of our answers, it's going to be 3.62 multiplied by 10 to the third meters per second. In other words, the speed of our debris is going to be equal to 3.62 multiplied by 10 to the third meters per second. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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