Hey, guys. So in this video, let's talk about the motion of satellites. Alright. So the first question is what exactly is a satellite? Well, a satellite is defined as any object that orbits another. So the idea is that if you have a large mass and a smaller mass that is going around it in, you know, some kind of shape, then that thing is a satellite. We have a couple of examples of this in real life. So the moon travels around the Earth. In that example, that's the little m, whereas the Earth is that thing in the center, the big M, and the Earth also travels around the sun. So in that situation, the Earth is actually the little m and the thing in the center, the middle, is the sun, that big M. Alright.

So, in satellite motion questions, you're often going to need to know what the shape of the orbit that that satellite makes. The shape of that orbit depends on 2 things. It depends on the object's speed and also its distance. So here's how this stuff came about. Right? Newton was thinking one day, what if you were to build like a huge cannon on top of a large tower off the earth and start to fire projectiles off of it? So I've got a little, like, sort of like velocity timeline here. That's going to show us what's going to happen as we vary the velocity. So first things first, if you were to go if you were to just drop this thing with a velocity of 0, then obviously it would just fall towards the center of the earth because gravity is pulling on it. Right? Well, what if you were to give this thing some initial velocity if you were to toss this or fire some cannon off of this? We have the object that has some horizontal velocity, but gravity still pulls on it. And so, eventually, it's just going to come and hit the earth.

If you were to throw this thing a little bit faster, this thing would travel farther around the earth because you start having some curvature here. But eventually, this force of gravity still means that the object hits the ground. And so we know how to deal with this stuff. This is just projectile motion, and we've dealt with this stuff before. Alright. Well, Newton had this idea that if you could throw an object fast enough, there is a minimum speed at which it'll just barely scrape the surface and eventually come back to where you started from. And that's the minimum speed required for an orbit. So the idea is that you're just barely scraping the surface of the earth, and the earth is still pulling on you so you're still falling. But you're traveling so fast that the earth is basically curving beneath you as you fall around it.

So that is the minimum speed required for an orbit. And so if there's a whole wide range of speeds that you can have that will give you an orbit. The idea is that there is a specific speed for which your orbit will be perfectly circular. That is the v circular, and you'll have a perfectly circular orbit. If you don't have this velocity, if you're somewhere around it, then your orbit is going to be elliptical. So that's why this v circular is just like a very specific point, whereas all of these values here are elliptical. Alright. So Newton had the idea of, okay, well, what if you throw this thing even faster than that, even faster than elliptical orbit?

Well, he thought that there was eventually a minimum speed for which this object will escape and so it'll never return. So the idea is that as you throw this thing farther, the force of gravity gets weaker, and eventually it gets so weak that this object will never return, and that is the escape velocity. Okay. So we said that the shape of the object depends on the speed, which we looked at here, but it also depends on the height. So what I want you guys to remember is that these values that I've made here, if I were to actually give them numbers, will change as your height changes. So for instance, if you were to build this tower even taller or shorter, then all of these velocities will actually change slightly.

Okay. And the last thing is that you're always going to assume that when you're working with satellite motion problems, you're going to assume that these are circular orbits just because the equations are going to be simpler, unless you're specifically told that it's not circular. Okay. So I want to put some numbers to these kinds of, to this kind of diagram here. So let's go ahead and check out this example. This is going to be mostly like a conceptual example. So we're standing on a tower, on some mysterious planet. And so from the height of that tower, the minimum speed to go around the planet without crashing is 2,000 meters per second.

We said that the minimum speed that you're just barely scraping the surface of that planet is going to be that v orbit. So that means that that is equal to 2,000 meters per second. Then we're told that the circular orbit speed is 5,000, and the escape velocity is equal to or the escape speed is equal to 10,000. Okay. So the idea is that we're going to be given these 4 speeds, and we're going to have to figure out just sort of conceptually what's going on here.

Okay. So, first things first is A. So A is saying what if we were to launch something at 1500 meters per second? Well, on the time line, that's going to be less than this 2,000 here, so it's going to fall somewhere around here. So we know that those things are going to be projectiles. So that means that a is going to be a projectile.

Alright. So what does b say? B is 4000 meters per second. So we're past this sort of threshold for the minimum orbit speed, but we're also less than this circular velocity. So anything in this green line here is going to be an elliptical orbit, remember. So that means that this is going to be elliptical. So we've got elliptical over here.

So what is c? So c says we have 6,000 meters per second. Well, we're not quite yet past the 10,000 meters per second escape speed. We're going to be somewhere over here. So this object is also going to have an elliptical orbit. It's going to be a slightly bigger elliptical orbit, depending on this faster velocity, but it's still going to be elliptical.

And lastly, if you have a launch speed that's 15,000 meters per second, now I have to ask if that's greater than the escape speed, and it is. So it's going to be somewhere over here. So remember that is the escape speed. So that means that d, this object has an escape. It's not really called an orbit. It's just actually just escaping. That would be the shape of the orbit that it makes. Alright, guys. Let me know if you have any questions with this stuff.