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Space Station Speed in Orbit

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Space Station Speed in Orbit : Speed necessary for the space station to stay in orbit

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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

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Space Station Speed in Orbit

Space Station Speed in Orbit : Speed necessary for the space station to stay in orbit