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- 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 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 km
- So we know that the magnitude of that centripetal acceleration has to be equal to this
- speed or the magnitude of the velocity squared divided by r
- where r is the radius of the circular path
- the radius of the orbit, which would be the radius of Earth plus the altitude
- So that we figured out in the last video, is 6771 km
- Let's just solve this for v. Then we can put in the numbers in our calculator
- So you multiply both sides by r and 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 for acceleration times the radius
- 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 is meter squared per second squared
- take the sqrt of that, you get meters per second
- which is the appropriate units
- So let's get our calculator out and actually calculate this
- There you go
- And then we want to calculate the principal square root of
- the magnitude of acceleration due to gravity at this altitude is 8.69 m/s squared
- times the radius of our circular path which is the radius of Earth which is 6371 km
- plus the 400 km of altitude that we have in this scenario
- So that gives us--we did this in the last video--6.771 times 10^6 m
- 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 our drumroll for how fast, in meters per second
- We get 7670
- 7671, but I'll just stick to three significant digits, 7670 meters per second
- Let me write that down
- The necessary velocity to stay in orbit is 7670 meters per second
- Let's just conceptualize that for a second
- Every second it's going over 7000 m, or every second it's going over 7 km
- If we assume that's the direction it's traveling, it's going at a super super fast speed
- If we want to translate that into kilometers per hour
- you just take 7670 meters per second
- and say, there are 3600 seconds per hour
- If you multiply that, that's how many meters it will travel in an hour
- If you want that in kilometers, you just divide it by 1000
- You have one kilometer for every 1000 m. Meters will cancel out
- Seconds will cancel out. And you're left with kilometers per hour. So let's do that
- So we multiply our previous answer by 3600 and then divide by 1000
- or you just multiply it by 3.6
- and then we get roughly 27,600 km/h
- So this is really an unfathomable speed
- And you might be wondering, how does such a big thing maintain that type of speed
- Even a jet plane which is nowhere near this fast has to have 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, whatever--
- let's focus on a jet plane. A jet plane has to travel through the air
- It actually uses the air as its propulsion, sucks in the air and spits out the air really fast
- But it has all of his air resistance
- So if the engines were to 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 you 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 you did not have gravity which is causing the centripetal acceleration
- it would just go in a straight path forever and ever
- And that brings up an interesting point because if you are in order 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
- 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