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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