Hey, guys. So now that we've talked about energy conservation for circular orbits, we're going to be talking about elliptical orbits in this video. There are a couple of really important things that you need to know, so let's go ahead and check it out. The first is that, unlike for circular orbits, the kinetic and the potential energies are not going to be constant. They're going to be changing. Let's go ahead and see why. Now, we said for circular orbits, all we needed was the distance little r, and we need the velocity at this point, and we can relate that to the total amount of energy. The kinetic energy was related to the velocity, and the potential energy was related to the distance. Well, the idea is that for elliptical orbits, what happens is this distance of little r is actually constantly changing throughout an orbit. So what happens is the energies are going to change.

Let's take a look at a spacecraft that's at the very, very farthest point in its orbit. Now, just remember that this farthest point over here is called the apoapsis distance and that's the farthest. Right? So, it has some velocity at this point, and gravity is going to pull it in throughout its orbit. And as it does that, it's going to speed up and accelerate. So then it gets to this point over here, which is the periapsis distance that is the closest that it gets, and it has some really really high velocity because gravity has pulled it in. So it's at maximum speed. Then as it goes out, gravity slows it down, so that means the velocity when it gets out here is going to be minimum. So, if it's at the closest distance of periapsis, which is over here, then that means that the speed is maximum. And if the speed is maximum, then that means that kinetic energy is maximum over here. Now, because this energy is still conserved. Right? Because we're still talking about gravity, if the kinetic energy goes up, the potential energy has to go down in order to keep it balanced. So that means that *U* is minimum here.

Well, then it gets to the other side of the orbit, so it gets to its apoapsis distance over here. And we know that the velocity is minimum over here. So if the speed is minimum, that means the kinetic energy is minimum. And, again, because the total amount of energy is conserved, as *K* goes down, *U* has to go up. So the potential energy is maximum here at apoapsis. Alright. So that's one really important point. So, basically, to wrap it up, the total amount of energy in circular orbits is constant and *K* and *U* are all constant. But for an elliptical orbit, the total amount of energy is constant, but between *K* and *U*, it is constantly changing. So what happens is one goes up, the other one goes down and vice versa.

Now the second most the second really important thing or the most important thing you need to know is how to compare the velocities at 2 different points in an elliptical orbit. So the idea is over here at any point, you can actually compare the velocity to between these two points over here. So if all we need are the distances and the speed. So if we have *r* so this is *v₁* and *r₁*, and we have *v₂* and *r₂*, we can basically draw a relationship between these two velocities and distances by using this really important equation, v1 r1 = v2 r2. So what happens is in problems, you're usually going to be given 3 out of these 4 variables, and you can use this equation to solve for the 4th one. Obviously, this is just an example, but you can solve for any one of those unknown variables. Now, this equation actually comes from a different conservation law that we haven't talked about in a while, which is conservation of angular momentum around any object is equal to *mvr*. So, if we're talking about an elliptical orbit, we can use conservation of angular momentum to compare any two points in that orbit. But because we're talking about the same mass, this little *m* will cancel, and you will just be left with this relationship over here. Alright? So that's basically it. Let's go ahead and see how we use it in an example.

So we've got a planet that is observed orbiting a distant star. The closest distance over here is going to be, remember, that's periapsis, and its velocity is given over here. So that's the velocity of periapsis. So now we're supposed to find out what is the velocity at its farthest distance, and this is a different distance over here, which is equal to *r _{a}*. That's apoapsis. Right? Remember that's the farthest distance. So we've got that we're comparing 2 different speeds and velocities, and we're looking for, in this case, we're looking for

*v*. So what this means is that these two points so we have

_{2}*v*is equal to the periapsis distance, and this

_{1}*r*over here is equal to

_{1}*r*, and this is equal to

_{p}*v*times

_{a}*r*. So in this case, it happens to be the apoapsis and the periapsis, but it doesn't necessarily have to always be those 2. So we actually have what 3 of these variables are, and we can go ahead and solve for that third one, and it's as simple as that. So if we've got, we're just going to move this over to the other side, and we've got vprpra = va. So in other words, we have, the velocity, which is 59 kilometers per second, and 0.31 for the distance divided by 0.47. Now, some of you might be wondering why I've chosen to put in those units instead of having to convert them. And the reason is that when we're plugging in for

_{a}*v*and

*r*, they actually can be non SI as long as they're consistent. So as long as we use the same units for

*VP*and

*RP*, then or in the same units for

*RP*and

*RA*, then we'll just get the correct units. So these things just have to be consistent with each other in order for us to plug them in. Okay? So if you work this out, you're going to get a velocity of 38.9. And because of the units, we're going to get kilometers per second. So just make sure that you're able to do this with your professor. If you see this on a homework problem and you're told to put it in meters per second, you're going to have to convert it. Alright? Anyway, so that's it for this one. Let me know if you guys have any questions.