Energy of Elliptical Orbits - Video Tutorials & Practice Problems
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1
concept
Speed and Energy of Elliptical Orbits
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Hey guys. So now that we talk about energy conservation for circular orbits, we're gonna be talking about elliptical orbits. In this video. There's a couple of really important things that you're gonna 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 could 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 circular orbits, what ha or sorry elliptical orbits, what happens is this distance of little R is actually constantly changing throughout an orbit. So what happens is the E energies are going to change? So 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 perry. So is called the apple app 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 perry axis 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 the maximum speed then as it goes out, gravity slows it down. So that means that the velocity when it gets out here is going to be minimum. So if it's at the closest distance of persis, which is over here, then that means that the speed is maximum. And if the speed is maximum, then it means the 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 it goes up, the potential energy has to go down in order to keep it balanced. So that means that U is min here. All right. Well, then it gets to the other side of the orbit. So it gets to its apple apsis distance over here. And we know that the velocity is minimum over here. So if the speed is minimum, that means that the kinetic energy is minimum. And again, because the kinetic, 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 apple apsis. All right. 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 you 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, but the most important thing you need to know is how to compare the velocities at two 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 if this is V one and R one and we have V two and R two, we can basically draw a relationship between these two velocities and distances by using this really important equation V one, R one is equal to V two R two. So what happens is in problems? You're usually going to be given three out of these four variables and you can use this equation to solve for the fourth 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 the conservation of angular momentum. So we have that if an, if an object is traveling around another one, then all you need is the mass, the velocity and the distance and the 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. All right. So that's basically it. Let's go ahead and see how we use it in an example. So we've got a star that is absorbed uh that is observed orbiting a distant star, sorry a planet. And the closest distance over here is gonna be remember that's persis and its velocity is given over here. So that's the velocity of persis. So now we're supposed to find out what is the velocity at its farthest distance? And this is a different uh distance over here, which is equal to R A that's apple apsis, right? Remember that's the farthest distance. So we've got that we're comparing two different speeds and velocities. And we're looking for, in this case, we're looking for V two. So what this means is that these two points? So we have V one is equal to the uh persis distance. And this R one over here is equal to RP and this is equal to va times R A. So in this case, it happens to be the apple apsis and the persis, but it doesn't necessarily have to always be those two. So we actually have what three 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, uh we're just gonna move this over to the other side and we've got V pr P divided by R A is equal to the velocity at apple apsis. So in other words, we have uh the velocity which is 59 kilometers per 2nd and 0.31 for the distance divided by 0.47. Now, some of you might be wondering why I've chose to put in those units instead of having to convert them. And the idea is that when we're plugging in for V and R, they actually can be non si as long as they are consistent. So as long as we use the same units for VP and RP, then or in the same units for RP and R A, then we'll just get the correct units. So these, these things just have to be consistent with each other in order for us to plug them in. OK. So if you work this out, you are going to get a velocity of 38.9. And because of the units, we're gonna get kilometers per second. So just make sure that you're able to do this with your professor. Uh If you see this on a homework problem and you're told to put it in meters per second, you're gonna have to convert it. All right. Anyway, so that's it for this one. Let me know if you guys have any questions.
2
Problem
Problem
Pluto travels in a fairly elliptical orbit around the Sun. At its closest distance of km, its orbital speed is 6.12 km/s. At its farthest, its orbital speed reduces to just 3.71 km/s. How far is it from the Sun at this point (in km)?
A
km
B
km
C
km
D
km
3
example
Comet Traveling Around the Sun
Video duration:
7m
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All right guys, let's work this soon out together. So we have a comet that's traveling around the sun travels once every 2000 years. And we're told some information about its closest approach. And in part, a, we're supposed to figure out how far, what is its farthest distance from the sun. So you should recognize that as the apple apsis. Let's just go ahead and draw a quick little diagram. So we've got the sun right there and I'm gonna have this highly elliptical orbit like this. Now, if we're trying to figure out the farthest distance that corresponds to that length right there, that's apple apsis. And that's what I'm really looking for. Now, there's only two other pieces of information I know about this problem. The first is that the entire orbital period for the whole thing is equal to 2000 years. So I've got that and I got that the apo apsis distance, the closest that it gets to the sun, which is RP is just given by some number three times 10 to the eighth kilometer. So I've got that. So let's look through my equations. Now, you might be tempted to sort of use the equation which, which we compare two parts of an elliptical orbit. That's V one R one equals V two R two. The problem with this equation is that we, if you look through, we don't have any of the velocities and we're gonna be solving for one of the distances. So that gives us with three unknowns for this problem. There's no way we're gonna be able to use that. So it can't be this. We're gonna have to look for some other equations to solve for R A. So let's go ahead and look through my equations involving R A. So let's see, I've got this equation over here and I've got, let's say the T set doesn't work because there's no R A in there, the two equations relate the semi major axis and the eccentricity. Both of those values are unknown. So I can't use either of those. So it's gonna have to be this equation over here, which is that the semi major axis is equal to R A plus RP divided by two. Cool. So I've got what RP is and if I'm trying to figure out what the apple axis is, there's only one equation or one unknown I need to solve for. And that is the semi major axis. OK. So unfortunately, I'm gonna need another equation for this. So let me go ahead and just move this over here. So how do I figure out what the semi major axis is OK. Let's go back to our equations. Now, I can't go for this equation because this is the equation I'm just coming from. So obviously, it can't be that one. Again, I don't have any information about the eccentricities. So that means it only leaves me with one option. I'm gonna have to use Kepler's third law for elliptical orbits. Now, this is really useful because I'm actually told some information about the orbital period and that's the variable T. So I'm definitely gonna use this equation. So I've got TST squared is equal to uh four pi squared, a cubed divided by GM. So all we have to do is just rearrange everything to the other side. The GM go up, the four pi squared is gonna go down and I'm gonna get GMT squared divided by four pi squared is equal to a cubed. So now if I rearrange, I'm just gonna get that A is equal to the cube roots of that whole entire mess over there. So I've got 6.67 times 10 to the minus 11. Now, I've got two times 10 to the 30th for the mass of the sun. Now, I need T squared. All right. The problem is that the T I'm given is in years. It's not in seconds. So I'm gonna have to first convert. So we've got T is equal to 2000. Now, I've got, uh that's gonna be in years. So I have to multiply by 365 days per year to cancel that out. Now, I gotta multiply by 86,400 because that's how many seconds there are in one day. So like, let me, uh, so we write that out again. So we've got seconds per day. If you didn't know that off the top of your head, that's totally fine. But the years are gonna cancel with the years, the days are gonna cancel with the days. And then t is just gonna be 6.31 times 10 to the 10 seconds. That's how many years there are in 2000 or so. That's how many seconds there are in 2000 years. Great. So we're just gonna plug that into this equation over here, 6.31 times 10 to the 10th. And now we have to square that. Don't forget that. And now we have to divide this whole entire expression by four pi squared. Don't forget to put that in parentheses anyway. So if you plug all of this huge mess into your calculator, you're gonna get a semi major axis of 2.38 times 10 to the 13th. And that's gonna be in meters, right? So we can actually plug this all the way back into our initial equation and then figure out what R A is. So let's just go ahead and do that really quick if we were trying to figure out R A is by isolating it. So let's see, we're gonna get that uh two A right? We're gonna multiply this up and then we're gonna have to subtract the RP over. So two A minus RP is going to be R A, right? That's gonna be our expression and that just comes directly from this equation right here. Great. So we've got R A is just gonna be, um let's see, we've got R A is gonna be two times, we've got 2.38 times 10 to the 13th minus the persis distance, which is in kilometers. So you have to be very careful because this unit right here is given in km. So this is actually three times 10, the 11th in meters, right? So this has to be in meters and this has to be in meters. So that means that our apple apps is distance over here. What I get is I get 4.73 times 10 to the 13th in meters and that's the answer to part A. So it goes very, very, very far out and then it comes in close. Usually that's what comets do. All right. So for part B now we're figuring out what is the ratio of the comet's speed at its closest point to its farthest point? What does that mean? We're looking for a ratio? So that means it's gonna be something divided by something else. Now, remember the velocity at its closest point is gonna be VP and its velocity at its farthest point is gonna be VA. So we're looking for this ratio over here now because we're comparing the speeds of two different points in the orbit. Now, we can actually use this formula over here. So we can use V one R, one is equal to V two R two. Now, we're comparing the apple apsis and the per axis. So we can say that this actually goes down to VA R A equals V PR P. And now what are we looking for? We're looking for? How do we set up this ratio of VP over R over VA? So what we have to do is we have to bring the VA over to the other side by dividing, but then we have to get rid of this VP or RP. So we have to move this to the other side. Now, what do you end up getting is that you get R A divided by RP is equal to VP divided by VA. So there is our target ratio. That's actually what we're looking for. And both of these numbers, the R A and the RP, we now both know the right, we have this equation over here uh for R A or this number. And we also have what RP is. So let's just go ahead and plug that in. So that means that this ratio is just gonna be 4.73 times 10 to the 13th divided by three times 10 to the 11th. Remember we have to convert that and make sure it's meters to meters. Just have to make sure that the right units. So what you're gonna get is you're gonna get a number that is 100 and 58. Now, this number is unit list because it's just a ratio. What this means is that the speed at the comet's closest point is 100 and 58 times faster than what it is at its farthest point. So this is just a dimensionless number. All right guys, that's it for this one. Let me know if you guys have any questions.
4
concept
Energy Conservation in Changing Elliptical Orbits
Video duration:
8m
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Hey guys. So now we know a little bit more about elliptical orbits. We're gonna see how we use energy conservation to transition from a circular to an elliptical orbit. Let's check it out. Now, the idea is that we're still changing from an orbit in this case, a circular to an elliptical one. So some work has to be done. And by the way, if you're going from an elliptical to a circular, still some work to be done. So everything I say can be done backwards here as well. Now we're still talking about changing orbits changing works. So we're gonna use energy conservation. Now, when we did this for circular orbits, we were able to combine these two kinetic and potential terms into a total amount of energy. So this simplified or energy conservation equation into the initial energy plus some work done is equal to the final energy. Now, the way this worked for circular orbits was that all we needed was this distance here of little R of R circular orbit. And we use this equation GMM over two R. Now we already know that this equation doesn't work because what happens is that as this orbit or as this object travels in its orbit, it goes from the per axis distance, which is the closest that it gets to the apo apsis, which is the farthest. So what happens is as it travels throughout its orbit, this distance of little R is constantly changing. All right. So we're gonna need a new equation to work with this. Now, how have we done this before? We've always taken an elliptical orbit and broken it up into its axis, the minor and the major axis. And we always said that the most important variable of an elliptical orbit is the semi major axis which is given by the letter A. Now this is not the same thing as the apple apsis, which is the farthest distance over here. So don't get those two things confused. But all we did was that anytime we had an equation that went from circular to elliptical, we just rep replaced the little R with A and it's gonna work the exact same way here. So we take this equation and we replace it with negative GMM over to A. So that means that these equations over here are gonna be the equations that you plug into your energy conservation here. You just have to make sure that you understand that you're going from a circular to an elliptical, which one is your initial and which one is your final and then plug in the appropriate expressions for those and to see how that works. We're gonna go ahead and do this quick example right over here. So you've got a spacecraft and we're told what the mass of this thing is. We're told it's initially in a circular orbit and we're gonna be calculating the energy required to change it to an elliptical orbit. And we're told some characteristics about that. So let's go ahead and just draw a quick little diagram right here. So I've got the earth and I've got the initial circular orbit and like that. Now let's try it again. There we go. That's a circular orbit. And then what happens is we're gonna transition to a elliptical orbit over here. So we have the initial distance of this circular orbit. That's little R and then we're told what the A PG distance is. Again, remember that is this farthest distance here, not the semi major axis. OK. So this is the apogee and we need to know how much work is done. So, or how much uh how much energy is required to do that. But it's also the same thing as the amount of work. So let's go ahead and set up our energy conservation equations. So we got initial energy plus the work is equal to the final energy. Now we're going from a circular orbit all the way to an elliptical orbit. So we just have to make sure the plug in the equations for those the total energy of a circular orbit is negative GMM over to R plus the work is equal to now negative GMM over to A. So we're, we're looking for the amount of work that's involved. So all I have to do is just take this term and move it to the other side and just have to, I have to uh plus it because I've got a minus sign right here. So I've got the work is equal to, I've got GMM over to R and it's positive minus GMM over to A. Now, let's look through, look through my variables, make sure I've got everything. So I've got all of these letters here are just constants. I can look them up on this table over here and I have the mass of the spacecraft. Now, I'm told what the R distance is. That's this 6.87 times 10 to the sixth. Now, the only other thing I'm told is the apogee distance. But I need a, I need to find out what little A is, that's the semi major axis. Remember that's the A is not the same thing as the apple axis distance. So I need another equation to figure out what A is. Let's go ahead and look at my equations over here. So I've got uh this T squared equals four pi A squared blah, blah. I've got Kepler's third law for elliptical orbits. Now, I've got A, I've got G and M, but I don't have any information about the period of the satellite. So I can't use that both of these equations over here involve a but they also related to the eccentricity of the orbit, which I also don't have. So I can't be using these equations either. So I'm gonna use this equation over here, which is my apple and Perry absence distance divided by two. So I've got R A plus RP divided by two. Now, the good thing is that I have the APO G distance. I'm just told what that is as a number over here. Now, what about this persis distance? What is that value? Well, let's take a look at my diagram, right? So I know that I've got the, I've got the semi major axis over here, which I'm trying to find. So that's gonna be this piece over here. So that's a and the apple or sorry, the peri axis, which is gonna be the closest distance is gonna be this piece over here. But notice how these two distances are actually the same. So this actually becomes a really, really important point that you need to know and that when you're going from a circular orbit like we did over here to an elliptical orbit, what happens is that your little R actually be comes RP or R A and it depends on the size of your new orbit. So I'm gonna give you a couple of rules to follow. Now, just like we did with this problem. If your new orbit is larger, then that means that your little R becomes RP, just like it did for this problem over here. Right. So these two things are the same exact variable. Now, if you knew or it's smaller, then it's the opposite. Your little R, which just becomes R A. So that means that we actually know what RP is, we can just go ahead and solve that. So R A is the A PG distance 2.0 times 10 to the seventh plus. Um And then we've got 6.87 times 10 to the sixth and we're gonna divide that by two. So that means that A is equal to 1.34 times 10 to the seventh. And that's meters. We're gonna plug that in there for our equation. So now we can figure out what the work is. The work is just equal to. Now, what we can do is we can take out this GMM over two as a common factor. So GMM over two, now we've got one over and the R is going to be 6.87 times 10 to the sixth minus one over. I've got 1.34 times 10 to the seventh. So if you work all of this out in your calculator, you're gonna get the work done is equal to 1.41 times 10 to the 11th in Jos. So this is our final answer for the amount of work. And because it's positive, that makes sense because we ended up with a larger orbit. So all of those rules for orbit still make sense uh in terms of work and positives and negatives. Now, the last thing I wanna mention is that we went in this example from a circular to an elliptical orbit. So you might be wondering what happens when you do it backwards. Well, when you're going from an elliptical to circular, all the, all the rules are backwards. So we've got this elliptical orbit over here and imagine we're going sort of in the, in the clockwise direction. Now, if we wanted a larger orbit, like for a, we'd actually have to do a burn over here. So we'd have to increase our velocity in order to basically circularize that orbit. So if you take a look at the diagram here in our new, in our old elliptical orbit, this distance over here is actually our apo apsis distance and that becomes the radius or the distance of our new circular orbit. So if you're going from an elliptical to a circular, your new orbit is larger, then that means that R A becomes your R and if it's smaller, then RP will become the new R. So you'd actually have to do a burn or over here, you'd have to swim delta V and this is your persis distance that would become your new orbital distance in a circular orbit. All right. So all the rules are just backwards. All right guys, that's it for this one. Let me know if you guys have any questions.
5
Problem
Problem
A comet is in a highly elliptical orbit around the Sun with a period of 2400 years. If this comet has a mass of approximately kg, what is the total energy of its orbit?
A
J
B
J
C
J
D
J
6
example
Changing from a Elliptical to Circular Orbit
Video duration:
6m
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Hey guys. So let's check out this problem. We've got a spacecraft that is traveling from one orbit to another, right? We got this 2000 kg spacecraft and we're being, we're being asked to find how much energy it takes in order to go from an orbit where it has a pair of some number and then an apogee of another number. So we're talking about some elliptical orbits right here. So let's go ahead and draw out this quick little diagram. So I've got the earth that's gonna be right over here and I've got some uh perigee and then I've got some apogee out here and I just have to connect those two things with an ellipse. So it's gonna look something like this. OK. So then what happens is that now it's gonna change to a circular orbit. But one of the things I noticed, one of the things that you should notice is that these actually are the same numbers right here. The apogee of the elliptical orbit. So that's right over here is actually the orbital radius of the circular orbit that it ends up in. So basically what happens is that it goes from this orbit of an elliptical. And then it basically just circularizes it to a circular orbit like that. So some of the words, this right here is our initial orbit and this over here is our final orbits. So we know we're gonna be dealing with changing orbits. We have to use energy conservation. So we're gonna go ahead and use our energy conservation equation. So we're being asked to find delta E, let's just go ahead and write out energy conservation. So for part A, we've got the kinetic and potential plus any work done equals final kinetic and potential. So what happens to this delta E? Where does this delta E come from? Well, remember that in energy conservation, this work is actually equal to the change in energy. So when they're asking for delta E, they're actually asking you to find what is the work that's done in order to change this initial elliptical orbit. So how much energy do you need to sort of bridge the gap between this elliptical orbit? Then you're gonna fire the thrusters of the engine or whatever and you're gonna end up in this final circular orbit over here? OK. So let's take a look at both sides of our energy conservation equation. We know that when we end up over here. So this is going to be a circular orbit whereas we're starting from an elliptical orbit. So let's take a look at our variables here. We've got some distances involved. And then we've got a final distance here. We're never told anything about velocities. So what we can do is we can use our shortcut equations for elliptical and circular orbit energies. So in other words, we can use these equations right here because they have the distance involved. So remember that this elliptical orbit equation just becomes negative G big M little M over two A where that's the semi major axis. then we've got the work that's done is equal to uh the circular energy. So it's gonna be negative GMM over two times R, right? So we just go ahead and say this is gonna be our circular orbit equation right here. And this is gonna be our elliptical orbit equation. We're just gonna use those two things. So that's where these equations come from. And we're being asked to find out what is the work that's done in order to do this. So if we go ahead and move everything over to the other side, I'm gonna move this term over. And what you should get is you should get the work that's done is equal to. Now what happens is this is gonna become positive when you move it to the other side. So we're gonna get GMM over to A minus GMM over to R. So we can pull out all of this stuff except for the A's and Rs as greatest common factors. And what we get is that the work that's done is equal to GMM over two. And then we've got one over a minus one over R. So now I can start plugging everything in assuming I have all the numbers. So I've got that, I'm going around the earth. So I have the mass of the earth and I have the gravitational constants. I have the mass of the spacecraft. And the problem is I don't actually have what the semi major axis is. So I never told that is specifically, I do know what this final position is. This final orbital distance. That's just this number here of 8.87. So I got that. So the, the only thing I have to do is just have to figure out what A is. So what are my equations for finding me now at the semi major axis? Well, hopefully you guys realize that we're given the information about the PG and the A PG. So we can actually use this equation right here for finding the semi major axis. So we've got R A plus RP divided by two. So in other words, I've got uh 6.87 times 10 to the sixth plus 8.87 times 10 to the sixth. You're gonna divide that by two and you're gonna get 7.87 divided uh times 10 to the six oops, not 16. So it's just like halfway in between these two numbers right there. OK. So now we're gonna take that and we're gonna plug it back into here. And so what happens is if you go ahead and plug in all of these numbers, we have all of these constants over here on the side, you've got the mass of the spacecraft and you just plug in these, uh the, you know, the, this number in here for A and we have this 8.87 for R, you should get the work that is done is equal to 5.7 times 10 to the ninth. And that's in Joules. So that's actually the answer for part A. So now we have to figure out in part B, what is this change in velocity required to actually do this? So in part B, what we're being asked to find is a change in velocity. In other words, delta V, now, how do we get this? How do we get what delta V is? Well, remember that if we're gonna be out here and we're gonna basically fire the thrusters, we have some delta V in order to circularize this orbit that energy or that change in velocity comes from the work that we're doing, the work that the engines are doing. So all we have to do is just say that the work which by the way is equal to the change in energy is just related to delta V by delta E equals one half M delta V squared. So we just need to figure out what delta V is by relating the mass of the spacecraft to the amount of change in energy that it requires. All right. So we're gonna go ahead and move this over to the other side. We've got the one half that goes over and the M that goes underneath. So we've got two times delta E divided by M is equal to delta V squared. So we just have to take the square roots and delta V is gonna be the square roots of two times 5.7 times 10 to the uh ninth. And then we've got divided by the mass of the spacecraft which is 2000. So if you work this out, you're gonna get 2400 m per second. It needs to increase its velocity by 2400 m per second in order to circularize that orbit. Notice also how we got a positive energy for work. Uh So the, this is a positive number and that makes sense because we're going from a smaller orbit out to a bigger orbit. We're increasing the size of that orbit. All right, guys, let me know if you guys have any questions with this.
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