 Lesson 7, orbital mechanics. So all we're going to be doing is trying to fine-tune a few specific examples and see if we can recognize what's going on in terms of work and energy and forces. So a mass little m, how about that little rocket, is moved from the surface of a planet to a distance big r from the planet's center. Lovely picture right there. Which of the following is true about the work done? Well, I would look at those answers. Here's what I would say. When we're talking on the cosmic scale, work is the change in potential plus the change in kinetic. But did this question say that we're in orbit or did it say lifted or moved or placed? I think looking at this diagram here, Meagan, this does not look like we're in orbit. It looks like we went straight out from the planet. So I don't think that we have any kinetic energy at the end. I don't think they've given it that tangent velocity. Otherwise, they would have drawn the rocket ship pointing like straight up and added a little dotted circle and said, hey, it's an orbit. That makes sense. So it's change in potential. What's change in anything, Meagan? Potential energy final minus potential energy initial. Can I use MGH? No. Why not? Well, on the cosmic scale, G is not 9.8. So we're gonna have to use the cosmic potential energy. The amount of work done is gonna be negative, big G, big M, little M over. Now, I was gonna write our final, but looking at my diagram, what symbol do they want me to use for the final radius? Capital R, okay. Minus negative, big G, big M, little M all over. And oh, I guess they want me to use little R for my initial. You may notice we have a minus minus, which is a plus. So what's the correct answer? Explain your answer. There. We've derived it. And we noticed last day in the day before you get some really big answers. It requires an awful lot of energy to get something not into orbit, just to get it into outer space so that if you let it go, it's gonna fall back down. Oh, and then you need to give it some kinetic energy to get it into orbit so that it can rotate around the planet as well. To escape from a planet's gravity means, in theory, to travel an infinite distance from the planet. In reality, it means to travel far enough away from the planet that the pull of gravity has become very small. To barely escape means to escape from a planet with no kinetic energy left at the end. So you end up at rest and at an infinite distance. Of course, it's possible to reach infinity and still be moving. But when we're talking about escape velocity, we did this last class, but just to jog your memory, we'll quickly rederive it. We said that if we're calculating escape velocity, it's actually a conservation of energy question. We said that when you're calculating escape velocity, it's kinetic energy initial, it's potential energy initial, it's kinetic energy final, it's potential energy final, plus heat, well, no, outer space, no heat, or very little. And we said, out at infinity, that's zero because it's relative to zero at infinity. And if we do the math just right, we come to a stop out at infinity. We're coasting, we're slowing down, we're slowing down, we're slowing down, we're slowing down, losing energy, losing kinetic, losing kinetic, and come to a stop right at infinity. Which gave us this, a half mv initial squared plus negative big g big m little m all over our initial equal zero. We then plus this guy over that side. And we said, if I want to calculate the escape velocity, it turns out to be a half mv, and instead of initial, I'll call it escape squared equals, oh, not negative anymore, Mr. Doot, because you've plussed it over big g big m little m all over r, not r squared r, by the way, just be really careful because sometimes kids, because they're used to the r squared from forces, want to put an r squared or make sure you know which equation you're using, I've tried to emphasize this unit, get your formula sheet out whatever possible, right? And Megan, we said, yay, the mass cancels, and we said, as it turns out, escape velocity ends up being two times 6.67 times 10 to the negative 11, times m all over the radius of the planet, square root of it. So you calculate the escape velocity of the sun, very big, tough to get away from the sun. Escape velocity of the earth was 11,000 dish, I think it was 1, 1, 2, 1, 1, 1, something, something, something, I can't remember how. It's about 11,000 meters per second, okay? So in example two, a 12,500 kilogram rocket blasts off from the earth, and it barely escapes the earth's gravity with no additional engine use. How fast was the rocket moving at launch? I think it's talking about escape velocity here. We could rederive it, kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final, but let's just jump down to, we have the equation right here. How fast was it moving? Two times 6.67 times 10 to the negative 11, mass of the earth 5.98 times 10 to the 24th, all over radius of the earth, 6.38 times 10 to the, what, I can never remember, 6? And I think, because this is escape velocity of the earth, we should get that 11,000 again, but just for calculator practice, let's see if we do. Do we? Where's your calculator? At home. Where's your cell phone? Power it off and come swap with me. Please. You don't hide it well, looking around sheepishly, not making any eye contact with it whatsoever. Where's your calculator? Why wouldn't it be here? Where's your cell phone? I'll swap truck keys. Although knowing you, I bet you this calculator might be worth more than the, well, okay. Oh no, it has a little remote thingy, which means it can't be that old. Oh, I'm sure. What do you get? 1.1 times 10 to the 4th, is that right? 1.12 times 10 to the 4th, if I recall the actual number, is that right? All right, hopefully you're getting more proficient at your calculator. Again, by the time the test rolls around, you better know how to type all this kind of stuff in, and if you can't, snag me after the lesson, I'll happily help you. I like number three, I like number three. And what I mean also is not just that each little parts of these is fair game on the test, there's going to be some neat, nerdy stuff that's going to come out here. So we have a stationary, you know what, Justin, I'm going to underline the word stationary, because I think that tells me that my initial speed is zero. Specifically, it tells me my initial kinetic energy is zero, okay. And we did talk about last day how that's in our magic physics world, that technically we're all on the earth and we're all moving in further south. If you are, the more kinetic energy you have from the rotation of the earth, which is why we try and launch stuff further south, you can use the earth's momentum to give yourself a bit of a boost, but we're keeping it simple. It blasts off from the earth, oh, and Megan, can you see the difference the way they drew the rocket ship this time? There, they're telling me it's in orbit. Oh, and it says circular orbit, 25,500 kilometers from the center of the planet. So they did give me a radius, it is from the center, but instead of meters, what did they give me? So I'm going to say this, the final radius is 25,500 and to go into meters, it's times by a thousand. It's going to be 2.55 times 10 to the 1, 2, 3, 4, 5, 6, 7 meters. Okay, A says write a work energy equation, work equals change in potential plus change in kinetic. And this time, none of these are zero because what's its final kinetic energy, zero, or is it in orbit? It's in orbit. B says, all right, let's find some stuff here. What's the first thing it asks me to find? Okay, speed, period, radius, forces. When do you use energy? When they start talking about work and they didn't in part I, it's orbital speed. Big G, big M, little M all over R squared equals M. Oh, which circular acceleration am I going to use here? So I got two equations. Yeah, they're talking speed, so I'll use the V squared over R. Wonderfully, the mass of my satellite cancels, does an R cancel? Yeah, this time because right Dylan, it would move up there and have one on top, two on the bottom. So one will. Oh, and I get this, V squared equals big G, big M over R or V equals the square root of big G, big M. Over R. I'm running out of room, so normally I would write the numbers out as well on a test. But since I see a bunch of questions in one page, I'm going to try and save writing. Let's see if we can go straight to our calculator here. Let's see. It's going to be 6.67 times 10 to the negative 11 times 5.98 times 10 to the 24th divided by the orbital radius 2.55 times 10 to the 7th square root of that, baby. 3,950 meters per second and that's the three sig figs, so I'm good with that. Is that okay so far? Okay, so that would be fair game as a part A. If I was doing this part B, I would probably ask for the period to find the period I would start here again, but instead of V squared over R, what would I write here if I wanted the period? 4 pi squared R over T squared and it's going to be a different cross-multiply equation. But since you have a T squared, you'll also end up square rooting. What if I ask you for the orbital radius? Now, I gave you the radius in this question, but what if instead I ask you for it, you'd still do this and then if I want the radius, I'm either going to be giving you the period or the velocity or some way to calculate one of those if I really want to get nasty. All right, part two, it's kinetic energy while in orbit. Well, kinetic energy is a half mv and I'm going to put that little orb there, V orbit squared. Now, I want to show you something kind of nerdly cool in terms of the algebra here. It's going to be a half times 12,500 times the number that's stored on my calculator, but look, look, look, look for just a second here. What is V squared algebraically? What's V squared the same as? Big G, big M over R. I want to plug that in for a second because I want to show you something kind of a neat little relationship as it turns out algebraically kinetic energy in orbit is a half little m, which I'm not going to write just yet. Big G, big M over R, I'm going to put the little m on that side because call me silly, isn't that potential energy, right? As it turns out, Justin, for what it's worth, it's just kind of an interesting, nerdy little fact. The kinetic energy of any satellite is half of its potential, but positive instead of negative. I don't use that very often. I don't memorize it, but it's kind of one of my built-in error checks. If I'm ever doing one of these and I notice, oh, that answer is half of the other one, I'm probably right. Meanwhile, we would go like this, 0.5 times little m is 12,500. Is that right? Times V squared is my answer squared. We would do that and I get 9.776. I'll go a few extra sig figs because it's not my final. Well, okay, 9.78 times 10 to the 10th. So, Conor, this is kind of interesting nerd trivia, but it is kind of neat that, oh, the kinetic is half the potential, but positive. What's the third thing they want me to find? You see, this is where if I was really, really clever, yeah, you know what? I would almost be so stunned here that I would drop something too, by the way, because this is going to be kind of neat. It wants the orbital potential energy now, Justin. I could, if I really want it to be sneaky, double the kinetic and put a negative in front of it. In other words, I'm pretty sure the answer is going to be this times negative 2. I bet you the answer is going to be that, don't write it down. I'll leave it on my calculator. Let's calculate it the long way because we need to practice. Orbital potential energy, so potential energy in orbit is going to be negative big g, big m, little m, all over our orbit. Where big g is, big m, mass of the Earth, little m, mass of the satellite, orbit 2.55 times 10 to the seventh. Again, in the interest of trying to save space here, I did this question with my block C's and I just ran out of room, so I'm trying to kind of write a bit less. I'm going to do something I normally would not do on a tester because I'm going to try going straight to my calculator. So you guys are okay on what all these values are. I'm going to try going negative 6.67 times 10 to the negative 11 times 5.98 times 10 to the 24th times 12,500 divided by 2.55. Come on. Times 10 to the seventh. If all goes well, ideally this should be twice the kinetic but negative. Oh, what did I forget? Oh, the seven. I hit it. Is it twice the kinetic but negative? Oh, you know what? All that means is I know I'm right. Which actually is kind of nice to know on a test. Negative 1.96 times 10 to the 11 joules. Why negative? Because where is zero out at infinity? What we're really saying is if you wanted to get to infinity, you'd have to add that much energy to the system. You'd have to burn that much fuel to get to zero. Total energy in orbit. I haven't taught this, but what do you think total energy would want us to do? What does the word total usually mean in terms of a mathematical operation? Add them up. Energy total is potential in orbit plus kinetic in orbit. Oh, hey, hey, hey, hey, hey, hey, hey, hey. We could type all this. Let's see if we can be clever. If this, let's try that again so it doesn't look negative. If this is half of this but positive, what will the total be? I think this but negative. Like I think, let's see, let's try it. That number plus 9.78 times 10 to the, what was it, 10th? Look at that. Basically the same number but negative. Negative 9.78 times 10 to the 10 joules. So there are some nice little mathematically nerdy relationships. Can you survive without knowing those, Kellan? Absolutely because I would have just got them on a test by typing in the numbers anyways. I probably on a test wouldn't have relied on these shortcuts because what if I get them mixed up, right? But I would certainly at the end kind of go, oh, look at that. I bet you I'm right because that's half of that and those two are the same. Good. V. It's potential energy at launch. I'll call that potential energy initial. That's going to be negative big G, big M, little M all over our Earth, right? Well, let's practice with our calculators one more time. What's its initial potential energy? And it's going to be negative because we haven't defined the Earth surface at zero. We've defined it better to be zero. It's going to be negative. You know what I'm going to do, boys and girls? I'm going to go second function, enter on my calculator until I get almost all of that stuff typed in. And I'm just going to change the bottom 6.38 times 10 to the 6. So that was clever of me. You get negative 7.81 times 10 to the 11. Okay. Learn how to use your science, especially the graphing calculators, but even the scientific ones will remember your last line. The graphing ones remember your last, I think 20 lines, which can be really handy. Negative 7.81 times 10 to the 11th joules. And the last thing, I think it's the last thing, the work. Now, I would have no problem instead of walking you through every one of these steps. In fact, I'm pretty sure on your test there is a question where I give you the introductory paragraph and I say, find the total work required to put it in orbit. And you have to clue in, well, work is change in potential plus change in kinetic. What's change in anything? So I would expect you, Ryan, to clue in. I have to find the change in kinetic. Initial is zero. Final. Oh, I have to find the orbital speed before I can actually find a half mb squared. Then I would expect you to clue in change in potential. Final minus initial. Oh, initial is the one on the earth. Final is the one in orbit, but I would probably do that on one line combined together. I don't know if I'd do it all separately. And then I'll expect you to clue in final. Well, let's write this out. Let's write this out. It's going to be potential energy final minus potential energy initial plus kinetic energy final. What happened to the minus kinetic energy initial? Zero. It starts at rest. All right. Have I got it all here? I think I do. If I just scroll up a tiny bit. So potential energy final, that's in orbit. That was negative negative. Here we go. Negative 1.96 times 10 to the 11th minus potential energy initial. Negative 7.81 times 10 to the 11th. And I notice I have a minus minus there, but I'll let my calculator take care of it. Plus kinetic energy final. 9.78 times 10 to the 10th. How much work would the rocket engines have to do to get in orbit? Minimum 6.83 times 10 to the 11th. Hopefully that kind of helps kind of clear up what you use, what you do, what you think where in each step. I hope, I hope, I hope. If not, sorry. Move that there. Next page. Example 4. It says, using the expression for orbital speed that V equals that, prove that in a circular orbit, kinetic energy is half the magnitude of the potential. Did we notice that on the previous question? The total orbit energy is half the potential. Did we notice that on the previous question? The total orbit energy is the opposite of the kinetic. Did we notice that on the previous question? Well, then I'm not going to do that. Turn the page. In every one of these, I'm going to talk about the most common one. Here is a bunch of different situations. So it says, write the work energy equations. A rocket at rest on the surface of the earth blasts off and rises to a maximum altitude. How much work is being done by the engines? So here you would go, work equals change in potential plus change in kinetic. Work equals potential final minus potential initial plus kinetic final minus kinetic initial. Oh, wait a minute, wait a minute. I think here we would say this. At rest and being lifted, we would say no, no. How do I know that has no kinetic energy at the end? If I wanted to, I would either tell you or if I wanted to suggest that it was an orbit, I'd tilt the rocket into a little circular path. A rocket moving at speed V at lift off near the earth surface rises to a height H with no engine use. Okay. This time, I think you would have this. Work is going to be potential energy final minus potential energy initial plus, now your kinetic final I think, because it says rises to a height, I think your final is still zero, but you'd still have, I'm going to write it in here, but you'd still have some initial kinetic energy because this time it says we're already moving near the earth surface. Right? What about a rocket at rest on the earth blasting off and barely escaping to infinity? Okay. This is escape velocity. Did we tackle that one with work? Did we tackle that one with conservation of energy? Remember, we did this. Kinetic energy initial plus potential energy initial equal nothing because we said the final was zero and the final was zero. And then we went a half and V initial squared plus negative big G big M a little over R. We were able to derive an expression for escape velocity. A rocket moving at speed V at lift off near the earth surface barely escapes with no engine use. You know what? I'm not going to worry about that one. That one's a little obscure. The one I really want to look at is E. A rocket at rest on the earth blasts off and enters a circular orbit. And then it says or radius R should be of radius R. Sorry, typo. That's going to be work equals change in potential plus change in kinetic where change in potential equals negative big G big M little M over R final minus negative big G big M little M over R initial. And where change in kinetic equals a half M V orbit squared. Remember how we found velocity or orbital velocity orbital speed? FG equals FC and away we go. A rocket moving at speed V at lift off rises with no further engine use and enters a circular orbit of radius R. I'm not going to worry about that one. And a satellite moving at speed V at a distance R from the earth's center falls to the earth's surface. I'm not really going to worry about that one. They're nearly cool, but this is what I'm going to hit ship with on the test. So a couple more questions and then technically you can now do every question on the review. Technically, except I do need to talk a little bit about some combined energy circular motion questions like the roller coaster and a few other ones. So for now though, homework number one, I think we can do number three, four, five is like that long one that we did there. Find the total orbit energy of the earth orbiting the sun. Rocket blasts off from the earth has found them seven I like. What do they want you to find in number seven? Yeah, for number seven, I would make a little note here. Use kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final. I think I would do that rather than using a work approach because this isn't talking about how much work it's saying. How high? What would your final kinetic be? Because it says before stopping. None of the rest of these will be zero. Make sure you use the cosmic potential energy. Eight is good. Same idea as number seven and 13. Oh, hey, let's do 14. Which of those is not a vector? Number 14, which of those is not a vector? Energy, right? There, hey, we just did 14 quickly. Just jogging our memories. Okay, so you got a good chunk, a fair bit of time here. I was going to go over your equilibrium momentum test. I will yet, but I want to actually give you a big chunk of class. No videos. I've showed you almost all of them where you can now work on homework, get caught up and start working on the big circular motion review. Okay.