05 - Intro to Conic Sections (Circles, Ellipses, Parabolas & Hyperbolas) - Graphing & More. - Free Educational videos for Students in K-12 | Lumos Learning

05 - Intro to Conic Sections (Circles, Ellipses, Parabolas & Hyperbolas) - Graphing & More. - Free Educational videos for Students in k-12


05 - Intro to Conic Sections (Circles, Ellipses, Parabolas & Hyperbolas) - Graphing & More. - By Math and Science



Transcript
00:00 Hello , Welcome back to algebra . The title of
00:03 this lesson is called Introduction and applications of Connick sections
00:07 were starting a new kind of topic area here with
00:10 an algebra one of the most important and it's called
00:12 Connick sections . And to be honest with you it
00:15 is rare that I get so excited than what I
00:17 am right now about to teach this because I would
00:20 like to to pull from two or three different areas
00:23 , you know , in my experience kind of learning
00:25 different things and try to mix it all together too
00:28 so that you understand what comic sections are , why
00:31 they are important and how they are used at least
00:33 one or two good examples of how they are used
00:36 because comic sections actually is used all around you and
00:39 they're really interesting to see how they kind of come
00:41 about . So um there's not gonna be much math
00:45 in here . In fact I think there's really no
00:46 math at all , there's no equations , I'm holding
00:48 the pen because I have to write a few things
00:49 down , but there's really not going to be much
00:51 math , it's just gonna be more learning by seeing
00:53 things . And then ultimately , as we go uh
00:56 down in the lessons , I will introduce the equations
00:59 for the different kinds of comic sections . All right
01:02 , so what is this thing called ? A comic
01:03 section anyway ? Right . Laconic section is basically a
01:07 class of shapes . And you've heard of all of
01:10 these shapes before , you've probably played with or seen
01:12 a lot of these shapes before . I know that
01:14 you have actually , but it turns out that you
01:16 can get all of these shapes by playing around with
01:20 these things called a cone . I know , I
01:22 know you've seen what a cone is . So ,
01:23 I have a couple of props here I made and
01:24 put together to show you where some of these things
01:27 come from . So , the four comic sections ,
01:29 there's four main shapes when we're going to talk about
01:31 where they come from , what the four shapes are
01:33 . And then stick with me because I'm going to
01:35 show you one of the most amazing applications of comic
01:38 sections , which is the orbit of the planets and
01:41 the asteroids . All of the orbits of everything in
01:43 space is all a different type of comic section .
01:45 So stick with me for that . All right ,
01:47 so there's four kinds of comic sections . The first
01:49 one , you know of , it's called a circle
01:51 . Then we have one that we've also talked about
01:53 quite a bit in algebra called a parabola . Right
01:56 . But they all come from this uh this cone
01:58 that we're going to talk about in just a second
02:01 . So we have circles , we have parabolas .
02:03 The other two are ones that we haven't really talked
02:05 about too much yet . One of them is called
02:07 the ellipse , but I know that you know what
02:09 an ellipse looks like , It's kind of like an
02:10 egg shape shape . And then we have the final
02:13 one which we've talked about briefly called the hyperbole .
02:16 So we have circles , we have parabolas , we
02:19 have ellipses and we have hyperbole is those are the
02:22 four shapes . Now they are different , they do
02:25 look different , but they're related to one another because
02:27 the most important feature of all of these shapes ,
02:29 these comic sections are is that we can get all
02:32 of these shapes by cutting and slicing a cone in
02:35 different ways and that's why they're all called connick sections
02:38 . That's something you don't often learn in algebra ,
02:40 but all of these shapes come from cutting a cone
02:43 up . So let's talk about the first , the
02:46 first shape . We're going to talk about a circle
02:48 . I know we all know what a circle looks
02:50 like , I mean with john circle , since we
02:51 were in in you know , young kids or whatever
02:54 . But here we have a cone . How can
02:57 we get a circle from this cone ? If you
02:59 can imagine taking a saw or a knife and slicing
03:04 through this cone , exactly perpendicular to the base ,
03:07 so you're cutting straight across like this , and if
03:10 you could kind of separate it and look at it
03:12 , then the cross section of what you would cut
03:14 through would be called a circle . And so I've
03:16 actually done this right here , so here's another one
03:18 of these cones and I do this myself . Okay
03:21 , so it's not perfect , but if you slice
03:23 through this cone , uh exactly kind of perpendicular or
03:26 parallel with the base , I should say , and
03:29 take the top off and look at what you get
03:31 . What you get actually is a circle . This
03:33 is a perfect circular figure . Now , if you
03:35 had a perfect cone and a perfect knife and you
03:37 do everything perfectly , this would be a perfect circular
03:39 shape . You can see the exact same shape on
03:41 the part that you've cut off as well , so
03:43 you can see that you can get this beautiful perfect
03:46 shape . You know the ancient Greeks ? Thought circles
03:48 were the most perfect shape . Right ? You get
03:50 this perfect shape called a circle , simply by slicing
03:53 through a cone in a certain way . Right ?
03:57 And so the shape of a circle you all know
04:00 is I'm gonna try not to screw it up too
04:02 bad , looks pretty much something like this . Is
04:04 that perfect ? No , it's not perfect , but
04:06 the shape of a circle can come from slicing through
04:09 a comb . Now the next shape is for the
04:13 next type of comic section beyond a circle is what
04:17 we call . Uh I kind of do them in
04:18 different orders but I like to do the next one
04:20 because it's mostly related closely related to a circle .
04:23 It's called an ellipse . Now I know that you
04:25 can probably have an idea of what your head ,
04:27 what an ellipse looks like . It's kind of like
04:29 an egg shaped . But in math we have more
04:32 formal definition . So if you want to take the
04:33 same comb instead of slicing through it exactly like this
04:37 , if you were to take pretend this piece of
04:39 paper is a salt . If I were to slice
04:41 through it at an angle . In other words don't
04:43 slice through it like this straight on . Don't slice
04:45 through it like this straight on but tilt it at
04:47 any angle I want . It doesn't matter what angle
04:50 . Just any angle . Just pick it and sliced
04:52 through it like that . And then look at the
04:54 cross section that's cut away from that . You would
04:56 see that it would form your typical egg shaped type
04:59 of ellipse . Alright , so here's my best representation
05:02 of that . This is perfect . No but here's
05:04 another one of these cones . And you can see
05:07 that I've cut through it at an angle . Is
05:09 that angle straight across ? No it's at some oblique
05:13 kind of angle like this . Just picked a random
05:15 angle . If you were to take apart the thing
05:17 and look at it , you would see that this
05:19 is a perfect oval shape . Is it perfect ?
05:22 No because I did this by hand and I colored
05:24 it by hand . But that's the idea . It
05:25 gives you an oval shape . And if you look
05:27 at the part that I cut off it's it's more
05:28 or less oval shaped as well . So I'm using
05:30 the word oval , but the proper mathematical word is
05:33 called an ellipse . This is a mathematical shape .
05:35 It's called a ellipse . It's part of the family
05:38 of what we call connick sections , because we can
05:41 take a section of a comic a cone and generate
05:44 that shape that we call an ellipse . So the
05:47 next comic section that we learn about is called an
05:51 ellipse . Right now , this ellipse . Uh you
05:56 all know more or less what in the lips looks
05:58 like , but it comes from slicing a cone in
06:00 a certain way . Now I'm probably not gonna draw
06:02 a great ellipse and I apologize for that . But
06:04 here's my best , my best effort . Okay ,
06:07 It's not perfect , it's a little bit too flat
06:09 . You can kind of see that I flattened it
06:11 too much . But the bottom line is it's an
06:13 oval shaped thing . Now , the exact size and
06:16 shape of the circle , the exact size and shape
06:18 of the ellipse . All depends on exactly how you
06:20 cut through the cone . I just picked a random
06:23 angle . If I pick a different angle , the
06:24 shape of the ellipse will be slightly different , but
06:26 it will have the overall same characteristics . All right
06:29 . The other thing is that the circle can have
06:32 different sizes . Okay , of course , and the
06:34 ellipse can have different that it can be really stretched
06:36 thin . It can be closer and closer to a
06:38 circle . All of those details come from exactly how
06:40 you cut the cone . And the equation of a
06:43 circle is something we're gonna learn very soon . The
06:45 equation of an ellipse will see it looks very similar
06:48 to the equation of a circle , and we'll see
06:50 how it describes each of these shapes . Now ,
06:52 the third Connick section is one that we've talked about
06:56 quite a bit before , because we've learned about parabolas
06:59 . Remember Parabolas ? Like why is equal to X
07:01 square ? That forms that nice smiley face . Parabolic
07:04 shape . The parabolas shape . It's something we've been
07:07 learning about for three or four units now , in
07:10 algebra called the parabola , it is also one of
07:13 the comic sections , and it can also be obtained
07:15 by taking one of these cones and slicing through it
07:18 . How do you make a parabola from a comb
07:20 ? Well , if you can imagine , instead of
07:22 slicing it , we're not gonna slice of horizontal ,
07:24 we're not gonna slice it at just some random angle
07:27 . What we're gonna do is if you notice the
07:28 cone has kind of the sides for have a nice
07:32 angle to it , right ? However , whatever your
07:33 cone looks like , it's going to have some kind
07:35 of angle if I cut through it and see if
07:37 I can gesture with my hand if I cut through
07:40 it parallel to this side . Like if I'm gonna
07:43 kind of mirror this side , I'm gonna take my
07:45 hand and I'm gonna cut parallel through the other side
07:48 of the cone , but parallel to this side .
07:50 In other words , not just any angle I want
07:52 , I'm gonna match it to this side and cut
07:54 straight through it . Then what I should get is
07:56 called a parabola . So it's the very special case
07:59 when I cut exactly parallel to the other side .
08:02 So this is my best effort at trying to do
08:05 that . Okay so you can see here is my
08:08 cone and you can see I have one side down
08:11 like this and you can kind of see the cut
08:13 that I made as best I could . I tried
08:15 to cut it as as parallel as I could straight
08:18 through like this . So I'm gonna separate that for
08:20 you and I'm gonna show that to you and you
08:22 can see that looks like a problem . In fact
08:24 let me go and put this down and let me
08:26 turn it upside down because that's usually how we graph
08:28 problems , we kinda oftentimes graph them upside down ,
08:30 but you all know that problems can be the other
08:32 way as well , so you can see it has
08:33 that perfect shape , it comes down , it's nice
08:35 and rounded at the bottom like this and then it
08:37 goes back up . This is called a parabola ,
08:40 so it comes from slicing a cone . So it
08:42 is one of the comic section . So now we
08:44 have circles which are cut by taking a cone and
08:47 cutting it straight across . We have ellipses which is
08:50 slicing a cone at some random angle . And then
08:53 we have parabolas which comes from slicing it through an
08:56 exact angle parallel to the other side of the cone
08:59 . And then we have the 4th Connick section ,
09:01 which is called a hyperbole . This is when we
09:03 haven't talked about very much at all , and of
09:05 course there's an equation of a circle and an equation
09:08 of the lips and there's a different equation for problems
09:10 , which we've actually looked at a lot . And
09:12 then we'll be talking about hyperbole is which will have
09:14 a special equation for them as well . So hyperbole
09:17 is probably the hardest one to describe . So I'll
09:20 do my best . You have to envision two cones
09:23 , one on top of the other like this ,
09:26 and if you do that , it kind of makes
09:28 sense because let me kind of , we're gonna do
09:30 it like this . I guess if you imagine two
09:33 cones each of these shapes so far . In fact
09:36 , I kind of forgot to write uh Parable it
09:39 down . Let me take a second to do that
09:41 . So let's write parabola . I apologize for that
09:46 . We've done Parabolas so much , I kind of
09:48 forgot to write it down . So the problem is
09:50 this nice rounded bottom shape that kind of goes up
09:53 like this , that's what a parable is . Um
09:55 But notice that we cut the cone in one location
09:58 , we got one shape , we cut in a
10:00 different place . We got one thing called in the
10:02 lips . We cut it parallel to the other side
10:04 . We got one shape called a parabola . But
10:06 if you really think about it , the all of
10:10 the shapes that we have done so far , really
10:13 , it's easier to visualize them if you visualize the
10:15 cones one on top of another , like this All
10:17 right ? So if we take the top cone and
10:20 just cut it in one location , we get one
10:22 circle only one shape . If we take the top
10:24 cone and cut it at some random angle , we
10:26 get one shape . It's called in the lips .
10:28 If we take the top cone and we cut it
10:30 exactly parallel to the other side , we again only
10:33 get one shape . It's called a parabola . But
10:35 now we're gonna take these two cones as a pair
10:38 and we're going to cut them . But instead we're
10:40 not gonna cut them and a random angle and we're
10:42 not going to cut them parallel to the side .
10:43 We're gonna cut them straight up and down . So
10:45 if you can imagine this piece of paper is a
10:47 saw . I'm going to slice through the top straight
10:50 up and down . And of course then I have
10:51 to slice through the bottom one as well . So
10:54 this shape because it's actually two cones kind of stacked
10:57 on top of each other , It slices through two
10:59 different cones . So a high Papua actually has kind
11:02 of like two curves . They kind of come in
11:04 pairs . So the parabola is only like one curve
11:07 that's generated , you can see how we did that
11:09 a minute ago . The ellipse in the in the
11:11 circle only generate one curve when we slice through .
11:14 But because we're slicing through two cones straight up and
11:17 down , a hyperbole kind of comes in two halves
11:20 to opposite lee oriented curves . And they look kind
11:23 of like parabolas , but they're not the same thing
11:25 as a parabola . So let me show you what
11:27 happens whenever we take a cone and we try to
11:31 slice it in that way to form a hyperbole .
11:34 So again you have to envision there's two of these
11:36 , one on top of another , I only cut
11:38 one of them so you take this guy and you
11:40 slice it not in a random angle , but straight
11:43 up and down straight . It's not parallel to this
11:45 , not parallel to this straight up and down and
11:47 you separate that and this is what you get ,
11:49 you get something that goes up , it's rounded and
11:51 it kind of comes back down like this . Now
11:53 again , if I had another cone on top of
11:56 this like this and I sliced through the bottom one
12:00 and then I also slice through the top one ,
12:02 I would have another kind of cousin curve kind of
12:05 coming from the top cone here . I just didn't
12:07 actually slice through two of them for the demo here
12:09 . But the point is hyperbole has come in pairs
12:12 so if I'm gonna write that down for you ,
12:16 hyperbole as there's many ways to draw it , so
12:23 I'm just gonna pick my way . But basically it
12:26 looks something like this , it kind of comes in
12:29 at kind of more of a point . Actually ,
12:32 this should be a little bit straighter and then it
12:36 kind of goes in like this now notice right away
12:42 that the parabola and hyperbole do look kind of similar
12:45 , but they are different because the parabola is more
12:48 rounded , it's rounded on the sides , more ,
12:50 it's rounded on the bottom , more , whereas the
12:52 hyperbole is rounded at the bottom , but it's a
12:54 sharper point . And also once it kind of flares
12:57 out , it kind of goes straight , it goes
12:58 pretty straight pretty fast . And you can kind of
13:01 see that in the diagrams we have here , in
13:04 the in the cutaways here , if we compare the
13:07 parable of this shape to the hyperbole of this shape
13:10 , you can see this shape is much more rounded
13:12 , even even at the sides , it's more rounded
13:14 . Whereas the hyperbole to kind of goes in a
13:16 straight line , almost a straight line anyway , once
13:18 it gets far away from the point . And also
13:20 the point itself is more pointy for lack of a
13:22 better word , it's more of a sharp thing .
13:24 So a lot of people think hyperbole is um parable
13:27 is are really the same shape , but they're not
13:28 the same shape and they have different equations as well
13:32 . So this has no math in it . There
13:34 is an equation for a circle . We're gonna have
13:36 a whole lesson on the equation of a circle .
13:38 There is an equation of an ellipse . There's an
13:41 entire section on figuring out the equation of an ellipse
13:44 . There are equations for parabolas , we've actually seen
13:47 them before . We're going to revisit them here as
13:48 well and there's an equation that describes these pair of
13:52 curbs that we call a hyperba . The purpose of
13:54 this lesson is not to teach you those equations right
13:56 now . I'm gonna teach you those equations in a
13:58 couple of lessons . The point is to tell you
14:00 that all of those shapes can come from slicing a
14:03 cone and that's actually kind of neat because why ,
14:06 why do we care about this ? Why did mathematicians
14:08 care about this ? Well , we obviously know circles
14:10 are important . I mean lots of things have circular
14:12 shapes , lots of things have elliptical shapes . But
14:15 what does a parabola really mean in real life ?
14:17 What does the hyperbole really mean in real life ?
14:19 So what I would like to do is give you
14:21 a concrete example of why circles ellipses , Parabolas and
14:25 hyperbole are uh all very , very important and very
14:30 , very useful . And I'm gonna give you only
14:31 one example . There's many , many , many examples
14:33 we're gonna talk about the orbit of the planets .
14:36 It was one of the pioneering efforts a couple 100
14:39 years ago , several 100 years ago to prove that
14:41 the shape of the orbits of the planets and all
14:43 the stars and everything else . They follow these comic
14:46 sections , sometimes circular , sometimes elliptical , sometimes parabolic
14:50 and sometimes hyperbolic . So what I want to do
14:52 is get rid of all the props and I'm gonna
14:54 sketch all of those for you on the board so
14:56 that you can see that something as important as the
14:58 orbit of the planets actually comes from comic sections .
15:02 All right now to understand something about orbits , I
15:04 have to give you a couple of numbers otherwise nothing
15:06 will make sense when you're in orbit around the planet
15:08 . you have a certain speed , gravity is always
15:11 pulling you down into the surface of the planet ,
15:13 but you're not hitting the surface because you're going sideways
15:16 around the planet at a really fast speed . And
15:19 it turns out that the shape of the orbit that
15:21 you're in depends on your speed . That's pretty much
15:23 all it depends on . Right ? So what I
15:26 want to talk to you about is I want to
15:29 talk to you about something called the escape speed .
15:32 So V Sub E is called the escape speed scape
15:38 speed and this promise is going to have application iconic
15:41 sections here in just a second is gonna be the
15:43 escape speed . And what that means is that's the
15:45 speed that your spaceship needs to get to in order
15:48 to completely break free of gravity and continue on into
15:51 the universe without ever coming back . If your velocity
15:54 is less than the escape speed , then you're never
15:56 going to leave the planet , you'll just go around
15:58 and around some sort of shape . But if your
16:00 speed is greater than the escape speed , you're never
16:02 coming back no matter what , because you've escaped ,
16:05 you have enough energy to get away from the gravity
16:07 of the planet . Alright , so to put some
16:10 concrete numbers on here um for Earth , you might
16:16 wonder what is the escape speed of Earth ? The
16:19 sub e to escape from the gravity of the earth
16:22 , you need about 11.2 km per second . Now
16:26 , when you think about it , that's pretty darn
16:28 fast . Another second goes by , that's another 11.2
16:31 kilometers , another second goes by another 11.2 kilometers bam
16:34 bam bam every second other 11 kilometers , that's really
16:37 , really fast . If you're going less than 11.2
16:39 kilometers , you're not leaving the planet , you're just
16:41 gonna come back some sort of way . But if
16:43 you're going 113 km/s , then you will never come
16:46 back to the planet . You'll just continue on into
16:48 space right now , that's for the Earth . You
16:51 might imagine that for the moon , you don't need
16:54 As much speed to escape the moon . So the
16:57 escape speed for the moon is 2.38 km /s ,
17:04 km/s . All right , so that makes sense because
17:07 the moon is smaller and there's not as much gravity
17:09 . Now , what about for jupiter ? What is
17:13 the escape speed for Jupiter ? You might guess it's
17:15 higher . What do you think ? It's maybe 15
17:17 , maybe 20 25 kilometers per second ? No ,
17:20 no , no , jupiter is much , much bigger
17:22 than earth . You need 59.5 kilometers per second to
17:27 escape the gravitational pull of jupiter and not come after
17:30 . It's only almost 60 kilometers every second . Think
17:33 about that , Another second , another second , another
17:36 second , another 60 km almost each time . And
17:39 if you're less than this , you're coming right back
17:41 to jupiter , it just may take a long time
17:44 now . This is the jupiter . What do you
17:45 think it's going to take to escape from the sun
17:47 if you really launch a spacecraft out and try to
17:49 get it away from the sun , do you think
17:51 it's going to be maybe like 80 or 90 kilometers
17:53 per second ? Maybe 100 maybe 150 kilometers per second
17:57 ? No , no , no , the sun is
17:59 much , much , much bigger than even jupiter .
18:01 You need 618 kilometers every single second to be able
18:06 to escape the sun . And the crazy thing about
18:09 it is , we have actually built spacecraft and they
18:12 are in space right now and they are going faster
18:15 than this so they will never ever come back to
18:17 our solar system . They're just going to escape the
18:20 neighborhood of the sun floating in space along a certain
18:22 direction forever . And so we have built spaceships that
18:25 have achieved beyond escape velocity and so you should look
18:28 those up . But anyway here's the idea we have
18:31 escape speed . Why why am I telling you all
18:33 this stuff ? Because the shape of the orbit that
18:35 you're in is going to be one of these comic
18:37 sections and it's going to depend on your speed right
18:40 now . There's one more thing you need to understand
18:42 before we can actually draw the picture there . We
18:44 also have another speed called V . C . And
18:47 this is called the circular warm it speed . This
18:56 is just the speed required to stay in an orbit
18:58 . So these were the escape speeds . This is
19:00 how much speed you need to escape completely . I
19:02 need to be able to tell you how much speed
19:05 you need just to stay in orbit . All right
19:08 . So for an example uh for ISS the international
19:14 space station or you can think of the space shuttle
19:16 or any other spacecraft was sent up there . The
19:19 speed that it's in is roughly I'm gonna put a
19:22 little squiggly here because it's not exact but it changes
19:25 a little bit but it's roughly 7.66 kilometers per second
19:29 . So you can see right now if the speed
19:31 of the space station is 7.6 kilometers per second .
19:34 That is well below Earth's escape speed . And that's
19:37 a good thing because the astronauts don't really want to
19:39 escape Earth . They want to go round and round
19:41 forever . So the speed is less than the escape
19:43 speed . So that's why we stay in a circular
19:45 orbit . If you have a speed less than this
19:48 , then you're going to spiral down and you're gonna
19:50 eventually hit the ground , right ? If you have
19:52 a speed greater than this , you might get farther
19:55 and farther away from the planet , but you're not
19:57 going to escape the planet unless your speed is actually
19:59 bigger than 11.2 km/s . Now , why am I
20:02 writing all this stuff ? Because it was a triumph
20:06 of science uh back in the days of kepler and
20:09 Newton to figure out what gravity is . And and
20:13 of course our ideas on what gravity is has changed
20:15 since then , because now we know it's a curvature
20:17 of space and time . But still the idea that
20:20 this marker coming down is the force of gravity and
20:24 that that force of gravity is the same force that
20:27 kind of holds the moon in the orbit and the
20:29 moon is kind of traveling around under the influence of
20:31 Earth's gravity . It was a really huge leap of
20:34 knowledge to see that the force of gravity that's holding
20:37 me to the planet is the same force holding the
20:39 moon and kind of like keeping it going in a
20:42 circle as well . And it turns out that the
20:44 shape of the orbits , the act the shapes a
20:47 long time ago were thought to just be circles .
20:49 People thought they were just circles because the circle is
20:51 a perfect shape . I mean when you look at
20:53 all of these shapes on the board , which one
20:55 looks the most perfect to you . I mean my
20:57 money's on the circle . It looks beautiful . It's
20:59 just perfectly . It just it's beautiful . So you
21:02 know , back in the days before science people thought
21:04 everything must move in a circle , Right ? But
21:06 then it was gradually learned that the planetary orbits really
21:10 aren't circular . They're kind of close but they're not
21:11 really circular . And it turns out that the shape
21:14 of anything in space orbiting a gravitational field like this
21:18 is either gonna be a circle , it's going to
21:20 be an ellipse . It's gonna be a parabola or
21:22 it's gonna be a hyperbole . And my goal on
21:25 the next board is to show you why that's the
21:27 case or how that is the case . So in
21:29 order for us to do that , I need to
21:32 draw the Earth . So this is my best representation
21:35 of the Earth . It's not a good representation ,
21:38 but that's what it is . So this is the
21:39 Earth . Yes . Right . This dot right here
21:43 and this Earth has a gravitational field around it .
21:45 It's invisible . It curved space and time . And
21:47 so everything falls towards uh into towards Earth right now
21:52 you have at your disposal spaceship and the spaceship for
21:56 lack of a better , you know , representation .
21:58 I'm gonna draw it like this . This is a
22:00 really weird , not so great looking spaceship but your
22:03 spaceship is obviously some distance away from Earth now .
22:05 If the spaceship wasn't moving at all , if it
22:07 was just sitting there and then I just like let
22:09 it go of course the spaceship is going to be
22:11 attracted and pulled in and hit the surface of the
22:13 planet . So people think in space there's no gravity
22:17 . That's not true at all . Gravity extends everywhere
22:19 around the planet . There's gravity acting on the space
22:22 station , right ? There's gravity acting in the region
22:25 of space around mars around jupiter around the sun ,
22:27 there's always gravity . The reason the thing doesn't come
22:30 down to the planet like this is because it's being
22:33 pulled down but it's also moving sideways really , really
22:36 fast . So if it's moving so fast sideways then
22:39 the forces going to curve its path . It's still
22:42 trying to pull it in . But if it's moving
22:43 so fast sideways it's just gonna end up curving the
22:45 path into this thing called an orbit , right ?
22:48 That's why orbits have this circular shape . But as
22:50 we're going to see in a second they're not always
22:52 circles . They're actually most often not circles . All
22:55 right . So if you're in a spaceship and if
22:58 you just let go it's gonna try to come in
23:00 like this . But let's say you fire your thrusters
23:01 because you really don't want to crash into the planet
23:03 like this , right ? If you just give like
23:06 a little bit of thrust , just not enough to
23:09 make it a circular orbit , but just a little
23:11 bit of thrust then what you're gonna end up with
23:13 is the shape of this thing is gonna look like
23:16 an ellipse . Now , the Earth is actually at
23:21 what we call one of the folks . I you
23:23 can think of the word focused , having a plural
23:25 called fosse the Earth . Is that one of the
23:28 folks I one of the focuses for lack of better
23:30 word of this ellipse . We're gonna talk a lot
23:32 about focus later on . Don't worry about it right
23:34 now , but the Earth is situated at the focus
23:37 of this ellipse . And so if you just send
23:39 this thing on with a little bit of thrust like
23:41 this , it's going to end up in an elliptical
23:43 orbit that's very small compared to everything else . So
23:47 this is going to be any lips and this happens
23:51 if your velocity of your rocket is less than the
23:53 circular speed . Remember I told you everything needs a
23:56 circular speed to stay in orbit . The ISS needs
23:59 7.66 kilometers per second just to stay in a circular
24:03 orbit . But if we fire this rocket with less
24:05 than that let's say we fire it with two kilometers
24:08 per second then it's not gonna quite make a circular
24:11 orbit but it'll still have an orbit but it'll look
24:13 like any lips . That's one of the comic sections
24:16 . Okay , so now let's say we stick a
24:18 bigger fuel tank on there and we um we kick
24:22 it up a notch and we actually uh put more
24:27 velocity entire ship . Let's say we put enough velocity
24:30 in it so that it for earth anyway it has
24:33 a velocity of 7.66 km/s . Then that means in
24:37 that case that's not going to be in the lips
24:38 anymore . I just told you it's a circular speed
24:40 . So it's gonna go up like this . I'll
24:43 see how I'm gonna do this . Yeah , something
24:45 like this . And it's gonna more or less make
24:48 a circle . Now . This pink , this pink
24:50 thing is not exactly a circle . I'm doing freehand
24:52 . Okay ? But you can see right here that
24:54 this is called , this is gonna be a circle
24:57 , right ? And this means that the velocity is
25:00 equal to the circular speed , right ? Of the
25:02 circular velocity . So again we have a circular orbit
25:05 speed . This is the number . And so if
25:07 we have a velocity bigger than this one in the
25:09 first case , but exactly equal to the circular speed
25:13 , it forms a different shape called a circle .
25:15 And that circle is the exact shape that comes about
25:19 when you give the velocity of your ship 7.66 in
25:22 this case km/s . Now , the really interesting stuff
25:26 happens with what happens if you kick your speed up
25:31 beyond this . See the circular speed with 7.66 kilometers
25:35 per second . But for Earth , the escape speed
25:38 is way a period 11 . What happens if you
25:40 give it something like 10 kilometers a second ? So
25:42 it's got more speed than a circle , but not
25:45 enough speed to escape . It's right in the intermediate
25:48 area . What's going to happen then ? It's got
25:50 more speed than a circle , but less speed than
25:53 it required to escape . Let's say . We give
25:55 our ship even more energy , more speed , so
25:58 it's going to come up something like this , it's
26:01 going to make an ellipse that's going to come out
26:03 even farther and it's gonna come back to the same
26:06 location . You can see the black , the black
26:09 curve is an ellipse . It looks very similar to
26:11 what we do on the board as any lips .
26:12 So we're gonna write down that this is an ellipse
26:16 and this is if the velocity is greater than the
26:19 circular velocity , you read it this way velocities greater
26:21 than the circular velocity , but less than the escape
26:24 velocity . So the velocity is bigger than the circular
26:27 speed , but less than the escape speed . Then
26:30 it you see everything that's opening up . First it
26:32 was a smaller lips , then it turned into a
26:34 circle with more speed . Then it turned into this
26:37 large lips with even more speed . And then we
26:40 said eventually there's a barrier here right here . Of
26:42 course if we go above this speed , we're going
26:45 to escape and we're never gonna come back , We're
26:47 never going to come back . If we have give
26:48 ourselves like 12.5 or 15 kilometers per second . But
26:52 what happens if I give my ship exactly 11.2 km/s
26:57 ? What's going to happen if I give it exactly
26:59 equal to the escape velocity , you can imagine giving
27:04 more and more speed , let's say I give more
27:06 speed than the black , then it's gonna be a
27:08 larger lips , I give more speed than that one
27:11 , it's gonna get even a larger and larger and
27:13 larger lips . Eventually I'm gonna get exactly equal to
27:16 the escape speed and then the ellipse is gonna get
27:18 so big that it technically does close . But the
27:23 other end of the ellipses infinity away . That's what
27:26 happens when you escape right on the edge of having
27:29 the escape velocity , your lips , your orbital ellipse
27:32 get so gigantic that the other side of the thing
27:35 is across the universe and you never actually come back
27:38 , technically you're on in the lips , but it
27:40 would take infinity years to come back . The ellipse
27:42 got bigger and bigger and bigger and bigger and bigger
27:44 and that shape is what we draw as a parabola
27:48 on the board , noticed this parabola kind of looks
27:50 like half of an ellipse , but it's open ended
27:53 and it never ever closes on itself . Right ,
27:56 so that's what's going to happen if you give yourself
27:58 exactly equal to the escape speed . Okay , so
28:02 let's kick ourselves up to the point where we have
28:04 something like that , it's going to look something ,
28:07 something kind of sort of like this . And then
28:10 on the other side it'll look something kind of sort
28:12 of like this , technically this will form a closed
28:16 loop technically , when I say that , is that
28:19 you get your itself exactly this escape velocity , it'll
28:22 close up on itself . But like I said ,
28:23 the other end of this ellipses infinity light years away
28:26 . It's really never gonna close on itself . If
28:28 you have just a tiny micro micro meter per second
28:32 less , then the escape speed , then it will
28:34 close on itself millions of light years away . But
28:36 when you're exactly at the escape speed , the thing
28:38 never really closes and you're on what we call a
28:40 parabolic trajectory , or we call just a parabola problem
28:46 , and that's when the velocity is equal to the
28:48 escape velocity . Now , you might have guessed if
28:54 you're in a circle with a certain speed , you
28:55 give yourself more speed , it turns into any lips
28:58 , and if you give yourself more and more and
28:59 more speed , the ellipse gets bigger and bigger and
29:01 bigger . Eventually , if you get to the escape
29:03 speed , it opens up into this parabolic trajectory .
29:06 Well , if you give yourself more than the escape
29:09 speed , in other words , the space probes that
29:11 we built that are actually going faster than escape speed
29:14 for the solar system . They're not on a parabola
29:16 anymore . Those trajectories turn into this one called hyperbole
29:20 or also called a hyperbolic trajectory . It's kind of
29:24 hard to draw it because basically the thing opens up
29:27 even more , but essentially it's gonna look like this
29:31 . You can say if you give yourself even more
29:32 speed , I didn't like the way that started even
29:35 more speed . It kind of opens up , it
29:38 doesn't even bend over that much . It opens up
29:40 more like this , something like this and then on
29:43 the bottom you can kind of see it coming down
29:46 something like this . So you can imagine this thing
29:48 going on and on and on forever . So this
29:50 is when you have a hyper Bella So I'm gonna
29:53 have to kind of draw , I guess down here
29:56 , hyper below . Yes , hyperbole is when the
30:01 velocity is greater than the escape velocity . And this
30:05 is the master diagram of why I wanted to draw
30:07 this this right here . I'm summarising , I'm summarising
30:11 years and years and years of work by Newton and
30:13 kepler and all of those people that figured out what
30:15 the orbits of the planets were , because they could
30:17 measure the position of mars in the sky . They
30:19 can measure the position of jupiter in the sky ,
30:21 they can see the sun in the sky . They
30:23 measured it for years and years and years and years
30:25 and tried to figure out what the shape of them
30:27 were and it was very difficult because everybody thought the
30:30 shape of these planets were circles . It turns out
30:33 that all of the planets are actually an elliptical orbits
30:36 . They have a little more energy in general ,
30:39 a little more energy than is required to just maintain
30:41 a circular orbit . But they're pretty close to circles
30:44 , their elliptical , but they're just stretched a little
30:46 bit there , stretched a little bit beyond the circle
30:48 into elliptical territory and the sun . Is that what
30:52 we call ? One of the focuses or one of
30:53 the folks I of that ellipse . Okay , so
30:57 to bring it home , when we have something like
31:00 this on the board , what you have is you
31:03 have a circle , right ? You have a circle
31:07 at low energy , you give yourself more energy and
31:09 it turns into an ellipse right ? And then you
31:14 give yourself more energy and it turns into a parabola
31:18 and then you give yourself more energy . It turns
31:20 into a hyperbole for a circle , it goes around
31:25 and around and around the same shape over and over
31:27 again for a lips it also goes around and around
31:30 , but in an oblong kind of shape for a
31:32 parabola , technically , it does technically come around ,
31:34 but it's infinity years . So it really never comes
31:37 around . So it's exactly on that boundary between the
31:40 lips and hyperbole , right at the escape speed .
31:42 That's what a parabola is . And when you go
31:44 even faster than that , the thing opens up even
31:47 more . And so it's a hyperbole or it's one
31:51 half of that shape that we call the hyperbole .
31:52 So I'm gonna leave that on the board that's important
31:56 . And I want to close this lesson out by
31:58 talking specifically about Parabolas real quickly , because parameters are
32:01 one of the most important shapes that we have ,
32:03 all of these are important . But Parabolas , you've
32:06 seen Parable is a lot more than you realize .
32:08 And that's because Parabolas have a special property . Parabolas
32:15 have a very special property . That's very , very
32:17 , very , very useful . And we've used this
32:19 property a lot . So let me try to draw
32:22 a basic parabolic shape , something like this . Is
32:24 this perfect ? No , it's not perfect , but
32:26 it's generally pretty close now , let's say I'm having
32:30 gotten to it yet , but every parable has a
32:32 focus , right ? We're going to talk about what
32:34 that focus means in just a minute . What the
32:38 interesting thing about a parabola is if I shoot a
32:40 light ray straight down into this Parabola and it hits
32:44 , let's say I make the parable out of a
32:46 shiny reflective surface , it's gonna bounce off and it's
32:49 gonna go right into the focus . Now , if
32:51 I take another light ray , and again it goes
32:55 straight down . But this time at the bottom of
32:57 the problem , you see how the shape of it
32:58 is just perfect . So it's going to bounce that
33:00 thing off again straight into the focus , no matter
33:04 where I draw my light ray coming in , as
33:06 long as it comes down like this , then it's
33:08 gonna bounce right off of this thing and go straight
33:10 into the focus . Like this , no matter where
33:13 I pick , if I pick up a light ray
33:14 straight down here , it's going to bounce up and
33:15 go straight into the focus so you can see that
33:18 your headlights on your car are parabolas , you put
33:22 the light bulb right here , it goes in reverse
33:24 coming in , bouncing to the focus . But if
33:26 you have some light source at the focus , it
33:29 can go and bounce the light out . So we
33:31 put curved reflectors on the lights in your car to
33:35 bounce the light straight ahead . But more important than
33:38 that is how we build satellite dishes . I'm a
33:41 terrible artist , but here's a satellite dish . This
33:44 is some , you know , pointed at the sky
33:46 , This is some giant side like this and we
33:48 built to talk to aliens or something like this .
33:50 And then at the focus , which is right here
33:52 , you build a receiver , a really sensitive receiver
33:55 and you have to build some kind of like structure
33:57 to hold it in place . But basically you build
34:00 a receiver right here . So what this means is
34:03 that all of the radio waves and and all that
34:06 stuff coming from space come and hit your receiver and
34:09 no matter where they I'm sorry , they hit the
34:11 dish in the back which is a parabola , a
34:13 shape of a parable . It's very carefully , this
34:15 is not just a circle , it's a parabola and
34:17 it comes in and bounces off and hits the receiver
34:19 , bounces off and hits the receiver , bounces off
34:21 and hits the receiver . And so you basically build
34:23 a giant bucket , collecting all the radio waves ,
34:26 all of the waves bounce off and go straight into
34:29 the receiver . And that is how we build these
34:31 enormous radio telescopes that can listen to the sound ,
34:36 to the radio waves coming from those space probes and
34:38 they can amplify them so that we can measure them
34:41 . Because , believe me , a space probe out
34:43 beyond Pluto has a very small transmitter and that energy
34:46 coming from that from that space probe way out there
34:48 is incredibly weak micro watts , Probably even less than
34:51 that . I have to look it up , but
34:52 it's very , very , very small amount of power
34:55 . So the only way to amplify it is to
34:56 build an enormous dish to collect it and to bounce
35:00 all of the energy into the receiver , which is
35:02 right here . Right ? And this shape does not
35:05 work for a circle . A circle doesn't do this
35:07 , A hyperbole doesn't do this , eh A and
35:11 the lips doesn't do this , only a parabola does
35:13 this . And we're gonna talk a little bit more
35:14 about that later . Mhm . So we've covered a
35:17 lot in this section . We said circles , ellipses
35:20 parabolas and hyperbole can all come from slicing connick sections
35:24 . That's why they're called comic sections . We talked
35:26 about escape speeds and all that stuff , but that
35:28 was just to motivate talking about this because when you
35:32 take the theory of gravity , Newton's theory of gravity
35:34 and you run it through the equations , you can
35:36 prove that all of the orbits follow the special shapes
35:39 called comic sections . A low amount of energy as
35:42 a smaller lips a little bit higher is called a
35:44 circle higher than that is called A larger lips .
35:47 More and more and more energy gives bigger and bigger
35:49 , bigger ellipses . Eventually you get to the escape
35:51 speed , in which case it's not an ellipse at
35:53 all anymore , It's called a parabola . And beyond
35:55 that is called a hyperbole . When you have escape
35:57 velocity , when you have velocity greater than the escape
36:00 velocity . Now , in the subsequent sections , we're
36:03 going to dive deeper into these comic sections , we're
36:05 gonna talk about what exactly are those shapes , How
36:07 do we construct them ? And then we're gonna talk
36:09 about the equations of circles and parabolas and ellipses and
36:12 hyperbole and we're gonna solve problems where we try to
36:16 write the equation down , figure out what the focus
36:18 is , figure out what the vertex is . Figure
36:20 out all these different things about these comic sections .
36:23 And I hope that by learning about orbits and learning
36:25 about transmitters and receivers and satellite , this is you
36:28 can see how important this kind of stuff is to
36:30 everyday science and technology . So follow me on to
36:32 the next lesson . We're going to dive a little
36:34 bit deeper into how we get these exact shapes and
36:37 talk about the math behind it in comic sections .
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