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 stay 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 you 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 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 that the magnitude of that centripetal acceleration has to be equal to the speed or the magnitude of the velocity squared divided by divided by R what R is the radius of the circular path so in this case would 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 6771 kilometers and we could actually let's just solve this for V and then we can put in the numbers in our calculator 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 our 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 unit's work out this is meters per second squared times meters which gives us meter squared per second squared take the square root of that you get meters per second which is the appropriate units but let's get our calculator out and actually calculate this let me just my calculator sitting on my other screen and there you go and then we want to calculate the principle square root of the acceleration due to gravity at this out to the magnitude of the acceleration due to gravity that altitude is eight point six nine eight point six nine meters per second square 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 six point and we did this in the last video six point seven seven one times ten 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 unit's don't do anything strange and then we get a drumroll 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 seven thousand six hundred and seventy I could say seventy one but actually I'm just going to stick to three significant digits 7670 meters per second so let me write that down the necessary velocity to stay in orbit is seven thousand six hundred and seventy meters per second so let's just conceptualize that for a second every second it's going over seven thousand meters or every second it's going over seven kilometers every second it's going to get the super if we assume that's the direction is traveling and it's going at the super super fast speed and if we want to translate that into km/h you just take seven thousand six hundred and seventy 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 there's 3,600 seconds per hour and so if you multiply that that's how far how many meters it will travel in an hour but if you want that in kilometers you just divide by a thousand you have one kilometer for every one thousand meters meters will cancel out seconds will cancel out and you are left with kilometers per hour so let's do that so we get so that was our previous answer we multiply by 3600 multiply by 3600 and then divide by a thousand so we really could have just multiplied by three point six and then we get twenty-seven thousand roughly 27 thousand six hundred kilometers per hour so this is really an unfathomable speed and you might be wondering how to such a big thing maintain that type of speed because you know we you know even a jet plane which is nowhere near this fast has to have these huge engines to maintain it's it's 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 or if I throw a ball or anything like that but a jet plane let's take 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 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 shut down all the air all the air would provide all it 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 it not 100% complete vacuum but almost complete vacuum so it has pretty much no air resistance very you know 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 in that it'll keep staying that speed in fact if it did not have gravity if it did not have gravity which is creating this causing the 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 the speed too much if you slow down if you slow down you will slowly spiral into the earth and if you speed up a lot beyond that speed you'll slowly spiral away from the earth you'll 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