A 2,000-kg spacecraft is blasting away from the surface of an unknown planet the same size as the Earth. At 1500km above the surface, an instrument onboard reads the gravitational force to be 18000 N. What is the planet's mass?

Hey, guys. Hopefully, you're able to tackle this one on your own. So we got a 2,000 kilogram spacecraft that is blasting away from the surface of a planet. So let's go ahead and draw out that planet and I've got the satellite right here. So I'm told that the mass of this satellite is equal to 2,000 kilograms and that's little m. Right? So at, 1500 kilometers above the surface of some unknown planet, but it's the same size as the earth. So I've got the distance between the center out to the surface, which is equal to that big radius, is just equal to r e, which I have, in this table right here, the radius of the earth. And I'm told that the distance above the surface is is 1500 kilometers. Remember that represents little h, and that's just equal to the height above the surface. So I've got 1500 kilometers. And at this distance, there is a some onboard instrument that re that reads the gravitational force and that force always points towards the center of mass. And that gravitational force, fg, is equal to 18,000 newtons. And I'm supposed to be figuring out what is the planet's mass. So in other words, how can I use this to figure out what big M is? That's my target variable. So if this is my target variable, we've got one equation to use, we've got the gravitational force. So let's start out with that equation. We've got fg equals gbigmlittlemoverlittlersquared. Remember we're working with, a planet and something else, not 2 point masses, so that's where we're going to use this this formula right here. So, I've got this as my target variable, this big m. And then just remember that this little r distance is always equal to the radius plus the height. So in any problem, if you're given what the height is above the surface Oops. If you're given the height and you're also told that the radius, you can figure out what that little r is. It's just the combination of those 2. So if we look through all of our variables here, I have what the force of gravity is. I know that g is just a constant big g. And then I've got little m and this little r distance I have with the radius and the height. So I can use that to figure out little r. So I basically got everything else in this problem, all I have to do is just solve for little m. Or sorry, for big m. So let's go ahead and do that. So if we start moving this equation around, I got the r squared that's gonna come up to the top, and then the g and the little m is gonna go towards the bottom. So I've got that f g times r squared divided by bigglittlem is equal to the mass of the planet. So I've got the force of gravity, which is 18,000. That's newtons. And I've got this little r distance here. Right? So I'm told that that is the radius, which is the radius of the Earth. That's 6.37 times 10 to 6th plus the 1500 kilometers above the surface. So just remember that this is in kilometers. You have to multiply by a1000 to get into meters. So what you're gonna get is 1.5 times 10 to the 6th. So that is 1500 kilometres and expressed in metres. So, yeah, you just gotta add those two numbers and you should get 7.87 times 10 to the 6th, and that's what we're gonna plug into our formula. So we've got 7.87 times 10 to the 6th. Just Just remember that you have to square it, right, because we have this little r squared in the numerator. And now we have to divide it by the gravitational constant, 6.67 times 10 to the minus 11 times the mass of our our spacecraft, which is 2,000. So if you go ahead and plug all of those numbers into your calculator carefully, you should get the mass of the planet, which is 8.36 times 10 to the 24, and that's in kilograms. And that's it for this one. Let me know if you guys have any questions.

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A 2,000-kg spacecraft is blasting away from the surface of an unknown planet the same size as the Earth. At 1500km above the surface, an instrument onboard reads the gravitational force to be 18000 N. What is the planet's mass?

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