Satellite Speed

by Patrick Ford
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Hey, guys. So remember when we were talking about satellite motion? We said that there was a specific velocity a satellite needed. In order to go in a perfectly circular orbit, we're gonna need to know how to use that formula and calculate that velocity. So let's go ahead and cover that in this video. So we have a satellite that's out of some distance away from the earth. That center of mass distance is little are now at that distance. There is a gravitational force that constantly tries to pull it back towards the earth. So the question is, why doesn't just come crashing into the surface? Remember that this asked. The satellite is actually falling towards the earth, but it has a tangential velocity that basically keeps it going in a circle. So this thing is constantly falling around the earth, and the earth is trying to pull it backwards. But that tangential velocity keeps it going Eso that the earth is constantly curving beneath it, okay, And so this for a satellite in circular orbit that gravitational forces, actually what's causing it to go in a circle so that gravitational force keeps the satellite going in uniform circular motion and the relationship between that speed, which is the orbital speed and the distance, which is that little are is the V sat equation and that is the square root of capital G Capital M over. Little are so just want to remind you guys that that capital M is actually the mass of the big planet that's it's going around, not the mass of satellite. And that little art is not the radius. It's the orbital distance. It's that little our distance away. So sometimes you're gonna need to know where that equation comes from, so I can actually go ahead and work it out for you really quickly. So how do we get the velocity from uniforms? Circular motion? Well, remember that this if this is the forces acting on it, it's going uniforms, circular motion. And it has a centripetal acceleration, so we can actually start from F equals M A. To get to this, we know that f equals m A. But all these forces air centripetal, so the sum of all centripetal forces equals m A C. Now we know the only force that's acting on this is the force of gravity and that is m the A C becomes V squared over R So this is where this velocity actually comes from. Now we know what this force of gravity is. It's just g look big and little em over r squared as just Newton's law of gravity. And that's equal to M v squared over r. So to figure out what this V squared is, let's go ahead and simplify this equation. So I've got a little M that appears on both sides so I can cancel that out and I got a little are that also appears on both sides. And so what I'm left with is I'm left with capital G Capital M over our equals the square. So if I take the square roots, I'm just gonna get to that, uh, GM over r. I'm gonna get that lease at equation. So again, that is the relationship between the orbital speed and the orbital distance, which is your little are away from the surface. So I want you guys to remember is that for every value of our there is an exact speed. So there is an exact V sat in order to keep this thing going in circular motion. So, for instance, this satellite out here that's at some distance are has an exact V in order to keep it going in a perfect circle. Anything mawr or less than that, it's not gonna be perfectly circular. And also, if I wanted to change this orbit, if I wanted to go out farther or if I wanted to push this thing in closer, my V sat would have to change in order to keep this thing traveling in a perfect circle. So that's what that means. Alright, guys, let's go ahead and work out an example for this with the International Space station. So were asked to find the heights of the International Space station which travels around the earth. And we're told that the orbital speed is m per second in a nearly circular orbit. So any time you see this word nearly circular, you're just gonna assume that they're talking about a circular orbit so you can use all these equations to do that. Okay, so what are we told? We're actually trying to figure out what the height of this thing is, but remember that whenever we're trying to find little h or big are we're always gonna find always gonna find little our first and then we can relate it using the r equals big r plus h formula. So first we have to find Little are So how am I going to do that? Which equation? Um, I'm gonna use I'm trying to find what little our is, and I'm on. Lee told what the velocity of the satellite is so I can use the V sat equation in order to relate those two things because those are the only two variables that pop up in that equation. So let's start from the V sat equation. So I've got visa equals square root of GM over R. So now I want to actually get to what this are is so that I could basically get to what h is. Um eso I just have to go ahead and isolate that. I've got this are that's trapped in the denominator here s so I can lift the square roots by taking the square both sides. So I got visa squared, equals G m over our And now if I want to get our by itself, I basically want this thing to come up and I want the V sat to come down. So these things were just gonna trade places. So I've got that r equals gm over v sat squared. Now I just have to make sure I have all of this number, right? I have G, I have m and I have the sat squared and that capital M, because I'm going around the earth is just the mass of the Earth, which I have this table right here. So plugging all that stuff in I get 6.67 times 10 to the minus 11. I've got the mass of the earth 5.97 times 10 to the 24th. Whoops time, Senator, 24th and then divided by 7670. Just make sure that you square that in the denominator and you should get 6. times 10 to the sixth, and that's in meters. But just remember that we're not quite done yet because that was we've just only solved. We've only solved for little art. Now we have to go and plug it back into this equation to solve for H, which is the height. So we have little r equals big R plus h. So if we want to find a church, we have to sit. If the isolate h so h is equal to little ar minus big are so I got 6.77 times 10 to the sixth minus. What's this? What's this? Big are well this big are if we're talking about the earth is just the radius of the earth. So we're gonna have a track that 6.37 times 10 to the sixth and we should get four times 10 to the fifth, which is about 400 kilometers. You can actually go ahead and google this. If you google the height of the International space station orbit, you'll find that it is about 400 kilometers. On average, it goes up and down a little bit, but that's pretty much what it is on average, which is pretty cool. So you can use, um, you know F equals Emma and some simple equations to figure out how high the international space station actually orbits around the earth, which is pretty cool. Alright, guys, let me know if you have any questions with this stuff