Video Transcript
Two spaceships are in deep space. The distance between the centers of mass of the two spaceships is 300 meters. And the force between them is 5.51 times 10 to the minus seven newtons. If one spaceship has a mass of 24,000 kilograms, what is the mass of the other spaceship? Use a value of 6.67 times 10 to the minus 11 meters cubed per kilogram-second squared for the universal gravitational constant. Give your answer to the nearest kilogram.
So here are our two spaceships. The distance between their centers of mass is given to us as 300 meters. And we’ll call that 𝑑. Let’s call the masses of the two spaceships 𝑚 one and 𝑚 two. And we’re told that 𝑚 one is equal to 24,000 kilograms. And the value we need to find is 𝑚 two, the mass of the second spaceship. We’re told that these spaceships are in deep space. This means there’s nothing else nearby. And so we only need to consider the force due to the mass of the two spaceships.
So now we need to recall the equation for gravitational force. The gravitational force 𝐹 is equal to the universal gravitational constant 𝐺 times the mass of the first object 𝑚 one times the mass of the second object 𝑚 two divided by the distance between them squared. Now, we want to rearrange this in terms of the mass of the second object 𝑚 two, since this is what we want to find. So we’re going to multiply both sides by distance squared. And that gives us distance squared times force is equal to 𝐺 times mass one times mass two. We can now divide both sides by 𝐺 and by mass one. And now we have that 𝑚 two is equal to distance squared times force divided by the universal gravitational constant times mass one.
Now, if we substitute in all our numbers, we have our distance of 300 meters squared times the force of 5.51 times 10 to the minus seven newtons, which was given to us in the problem statement, divided by the universal gravitational constant 𝐺 times the mass of the first spaceship 𝑚 one. And this comes to 30,978.26. Now, we’re asked to give this to the nearest kilogram. So that’s 30,978. Now we used SI units all the way through. That’s meters, newtons, kilograms, and meters cubed per kilogram-second squared. So this mass will be in the SI unit of mass, which is kilograms. And so the mass of the other spaceship is equal to 30,978 kilograms.