Space station speed in orbit

Now that we know the magnitude of the acceleration due to gravity at 400 kilometers above the surface of the earth, where the space station might hang out, what I want to do in this video is think about how fast does the space station need to be moving in order to keep missing the earth as it's falling, or another way to think about it, in order to stay in orbit, in order to maintain its circular motion around the earth. So we know from our studies of circular motion so far, what's keeping it going in circular motion, assuming that it has a constant speed, is some type of centripetal acceleration. And that centripetal acceleration is the acceleration due to gravity. And we figured out what it was at 400 kilometers. And so we know that that centripetal acceleration-- let me write it here in pink-- we know that the magnitude of that centripetal acceleration has to be equal to the speed or the magnitude of the velocity squared divided by divided r, where r is the radius of the circular path. So in this case, it'd be the radius of the orbit, which would be the radius of the Earth, plus the altitude. So that, we already figured out in the last video, is 6,771 kilometers. Let's just solve this for v, and then we can put in the numbers in our calculators. So you multiply both sides by r, and you flip the two sides. You get v squared is equal to the magnitude of our acceleration times the radius. The magnitude of our velocity or speed is equal to the square root of our acceleration times-- or the magnitude of our acceleration-- times the radius. And so let's get our calculator out, and you can verify that the units work out. This is meters per second squared times meters, which gives us meters squared per second squared. Take the square root of that, and you get meters per second, which is the appropriate units. But let's get our calculator out and actually calculate this. Let me see. My calculator's sitting on my other screen. There you go. And then we want to calculate the principal square root of the acceleration due to gravity at this altitude, the magnitude of the acceleration due to gravity at that altitude is 8.69 meters per second squared times the radius of our circular path. That's going to be the radius of Earth, which is 6,371 kilometers plus the 400 kilometers of altitude that we have in this scenario. So that gives us 6.-- and we did this in the last video-- 6.771 times 10 to the sixth meters. And it's important that everything here is in meters. Our acceleration is in meters per second squared. This right over here is in meters. So the units don't do anything strange. And then we get a drum roll for how fast-- and this is going to be in meters per second. We already thought about how the units will work out. We get 7,670-- I could say 71. But actually I'm just going to stick to three significant digits. 7,670 meters per second. So let me write that down. The necessary velocity to stay in orbit is 7,670 meters per second. So let's just conceptualize that for a second. Every second, it's going over 7,000 meters. Or every second, it's going over 7 kilometers. Every second. It's going at this super-- if we assume that's the direction it's traveling, it's going at this super-, superfast speed. And if we want to translate that into kilometers per hour, you just take 7,670 meters per second. If you want to know how many meters it's going to do in an hour, you just say, well, there's 3,600 seconds per hour. And so if you multiply that, that's how many meters it will travel in an hour. But if you want that in kilometers, you just divide by 1,000. You have 1 kilometer for every 1,000 meters. Meters will cancel out. Seconds will cancel out. And you are left with kilometers per hour. So let's do that. So that was our previous answer. We multiply by 3,600 and then divide by 1,000. So we really could have just multiplied by 3.6. And then we get 27,000, roughly 27,600 kilometers per hour. So this is really an unfathomable speed. And you might be wondering, how does such a big thing maintain that type of speed, because even a jet plane, which is nowhere near this fast, has to have these huge engines to maintain its speed. How does this thing maintain it? And the difference between this and a jet plane is that a jet or a car or if I throw a ball or anything like that-- but a jet plane. Let's focus on a jet plane so we don't have to worry about other things. A jet plane has to travel through the air. It has to travel through the air. And actually, it uses the air as kind of its form of propulsion. It sucks in the air, and then it spits out the air really fast. But it has all of this air resistance. So if the engines were to just shut down, all the air would bump into the plane and provide essentially friction to slow down the plane. What the space station or the space shuttle or something in space has going for itself is that it's traveling in an almost complete vacuum. Not 100% complete vacuum, but almost complete vacuum. So it has pretty much no air resistance, negligible air resistance to have to deal with. So we know from Newton's laws, an object in motion tends to stay in motion. So once this thing gets going, it doesn't have air to slow it down. It'll keep staying that speed. In fact, if it did not have gravity, which is causing this centripetal acceleration right over here, it would just go in a straight path. It would go in a straight path forever and ever. And that brings up an interesting point. Because if you are in orbit like this, traveling at this very, very, very fast speed, you have to make sure that you don't vary from this speed too much. If you slow down, you will slowly spiral into the Earth. And if you speed up a lot beyond that speed, you will slowly spiral away from the Earth. Because then the centripetal acceleration due to gravity won't be enough to keep you in a perfect circular path. So you really have to stay pretty close to that speed right there.