Two 65 kg astronauts leave earth in a spacecraft, sitting 1.0 m apart. How far are they from the center of the earth when the gravitational force between them is as strong as the gravitational force of the earth on one of the astronauts?

Video transcript

Hey everyone. So this problem is dealing with gravitational force. Let's see what they're asking us. Two satellites named Cosco and solar server. Each with a mass of 120 kg are launched into space and positioned 48 m apart from each other. At what distance from Earth center will the gravitational force between the two satellites be equal to the gravitational force exerted by earth on a single satellite. Our multiple choice answers here are a 1.7 times 10 to the 13 m. B 2.27 times 10 to the 16 m C 2.31 times 10 to the 16 m or D 1.27 times 10 to the 13 m. So the key to this problem is going to be recalling that our gravitational force equation is given by F equals G M one M two divided by R squared. And here we have two different scenarios. We have scenario A where the satellites are exerting a force on each other. So satellite to satellite and scenario B where we're looking at one satellite and the force from the earth. So we're going to look at the, we're looking for the distance from the earth center where those forces are equal. So we can set these, this force equation equal to each other for each of the scenarios. So for scenario A, we have G M one in scenario AM two in scenario A divided by R A squared and that equals she times M one in scenario B um two in scenario B divided by R be swear. So from the problem, we are told that we'll call it M one A and M two A are the same because both satellites are equal to the same weight and they're also equal to the satellite. In scenario B and M two B is equal to the mass of the earth. And that is a constant that we can recall is 5.97 times to the 24 kg radius. For scenario A, the R, the distance between those two masses and that was given as 48 years. And then we're solving for this, our be fun. So when we look at the terms that cancel and isolating for a sub B, we have R and B is equal to the square root, um M two B multiplied by R A squared divided by two K. I went, I note here that the mass of each satellite was also given in the problem that's 120 kg. So now we have everything we need to plug in to solve for this R B or the distance um between the earth and the satellite. And so we have 5.97 times 10 to the 24 kg multiplied by m squared, divided by 120 kg. Take the swear of that, that is equal to 1.7 times 10 to the 13 m. And so that is the answer to this problem. And when we look at our multiple choice answers that aligns with answer choice. A so A is the correct choice for this problem. That's all we have for this one, we'll see you in the next video.