Find speed of second satellite, given speed of first
by Patrick Ford
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Hey, guys, let's check out this problem here. We're told to satellites are orbiting this planet and were given a whole bunch of numbers. So let's go ahead and just start drawing a diagram. So my diagram here is gonna involve some planet and I have two satellites in circular orbits, so let me go. Oops. Let me go ahead and drop my circular orbits. I've got one right there and we got another one that's gonna be right there. Let's say cool. So now I've got a mass. I've got a satellite that's over here. I'm told the mass of that satellite is equal to 68 and I'm told that the orbital radius, the orbital radius is six times 10 to the eighth. Remember that is that little our distance. And that is the distance between the centers of mass. That's not big R. And that's not a church. This is our But because the first one, because I have two satellites, I'm gonna call this our one. And then I'm told that the velocity, the orbital velocity, which is that tangential velocity of you one is equal to 3000, right? Cool. So I have another satellite that is some other distance away. So I'm gonna call this m two. And I know the mass is 84 then I've got the orbital radius of that thing. So then I've got here. The orbital radius is equal. Thio are too. And I know what that number is, right? That's nine times 10 to the eighth. Now I'm asked to figure out what is the orbital velocity off the second satellite. So really V two is my target variable. So let's start there. So I've got V two as my target variable. So how do I get that? Remember, from our equations, we're gonna use the V sad equation. We're looking for orbital velocity. So I've got V two is just gonna be the square roots of g times, Big M. And because I'm looking for V two, I'm gonna be plugging in our too into this. So let's take a look. This is my target variable right here. Let's take a look. If I have everything else in the problem, G is just a constant. Then I've got the mass of the planets, but I don't have that variable, so I've got the radius of the orbital radius, but I don't have Big M. So how do we go about solving for that? Because I'm gonna need that. I've never told it in the equation. So let's go over here and try to solve for big M. Let me go ahead, scroll down. So I've got Big M equals something. How do we get big? We can use either Forces Look at our table of equations right here. We can either use forces or we can use accelerations or we can use satellite motion. But I'm not told anything about forces in the problem. I don't have any forces between two objects and I don't have any any accelerations here. I know that it's circular motion and I know that I'm working with satellites, So let's use the other V sat equation. So remember, we have two satellites here, so let's look at the other satellite. The first one V one equals square roots of G times. Big M, divided by little are one. Now the are one of the other satellite. So if I take a look at this, I have the V one. I have G is just a constant and I have our one, so I can actually go ahead and figure out what this Big M is. So I'm gonna go ahead and rearrange. I've got this thing in the square root so I have to get rid of that square root by squaring both sides. I've got V one squared equals g times Big M over our one. Now I'm just gonna move everything over and sulfur are on sulfur m So I've got the one squared are one divided by G equals m. So I actually have everything there that orbital velocity for that first one is 3000. Then I've got a square that the orbital radius of the first satellite, we're told is six times 10 to the eighth and then we're just divided by the gravitational constant. And if you do that, you should get 8.1 times 10 to the 25th, and that's in kilograms. So now that we have that mass right here, we can actually plug it back into this equation for and sulfur V two. So let's go ahead and do that. So I've got V two equals square roots. And then I've got 6.67 times 10 to the minus 11. Then I've got the massive this thing 8.1 times 10 to the 25th. Now I've got two divided by the Oriole radius of the second satellite, which I'm told is 9.8. Sorry, nine times 10 to the eighth, and that's in meters. So you go ahead and do that. You get the velocity is equal to 2450 m per second. So we've got that number right here. Notice that this number is actually lowered. By the way, this is the final answer. Notice that this number is actually lower than the orbital velocity of the of the first satellite. And that's because it's a farther distance away. So we can see from the V sad equation that if our orbital distance is farther than your velocity is going to be lower. So that actually just makes some sense. Let me know if you guys have any questions with this