notes from the artist - clowder
TRANSCRIPT
1
Notes from the artistHow does a child learn new words? Kids learn by osmosis. After hearing a word often enough, it just sinks in.
In the process of coloring these pages, concepts and words will sink in over time.
This book is for kids from kindergarten to college.
I’m hoping this book wll expose younger students to concepts they normally wouldn’t see until higher grades.And that it will give advanced students some new views of concepts they’re already familiar with.
Thank you to Steven Pietrobon for his many helpful comments. Also to Isaac Kuo for his suggestion.They’ve helped me make this a better book. Any mistakes in this book are my own.
Hollister (Hop) David
Table of ContentsTwo Kinds Of Rulers...................................................................................................Pages 2 & 3A Point And A LineParabola ............................................................................................................................................4Ellipse................................................................................................................................................5Hyperbola .........................................................................................................................................6Cut ConesParabola, Circle, Ellipse, Hyperbola ...........................................................................................7Cone Of Light: Circle, Ellipse, Parabola ....................................................................................8Cone Of Light: Hyperbola ............................................................................................................9Two PointsEllipse ..............................................................................................................................................10Hyperbola........................................................................................................................................11Orbits in SpaceKepler’s 1st Law ............................................................................................................................12Astronomical Units, Hohmann Transfer Orbit .....................................................................13Parts Of An Ellipse: a, b, & ea...................................................................................................14Eccentricity Ellipse......................................................................................................................15Kepler’s 2nd Law ...........................................................................................................................16Specific Angular Momentum Cross Product Of Position And Velocity Vectors ...........17To The Second Power -- Squaring............................................................................................18Squaring Decimal Fractions .......................................................................................................19To The Third Power -- Cubing ..................................................................................................20Kepler’s Third Law........................................................................................................................21Pythagorean Theorem.................................................................................................................22Pythagorean Triangles ................................................................................................................23Ellipse b, ea As Legs And a As Hypotenuse Of Right Triangle ........................................24Dandelin SpheresDandelin Spheres ....................................................................................................Pages 25 - 29Two Lines (Cartesian Coordinates)Parabola..........................................................................................................................................30Parabola, x And y Axis ................................................................................................................31Circle...............................................................................................................................................32Area Of A Circle..........................................................................................................................33Ellipse .............................................................................................................................................34Area Of An Ellipse ......................................................................................................................35VelocitiesV Infinity .......................................................................................................................................36V Infinity And Hohmann Transfers ........................................................................................37V Infinity, V Hyperbola, V Escape, V Circular ......................................................................38Parts Of A Hyperbola.................................................................................................................39Mu....................................................................................................................................................40
Any corrections, suggestions or comments,please contact me at [email protected]
Evenly spaced parallel linesmeasure distance from a line.
7
8
TWO KINDS
2
1
2
3
5
6
4
Evenly spaced concentric circlesmeasure distance from a point.
1
2
34
5
6
7
8
OF RULERS
3
For each point on a parabola,Distance to Focus Point = Distance to Directrix Line.
Eccentricity = 1.
11
2
3
4
5
6
2
3
4
5
2 23
4
5
66
3
4
5
6
DIRECTRIX LINE
FOCUSPOINT
PARABOLA
4
For each point on this ellipse,Distance to Focus Point = 1/2 Distance to Directrix Line
Eccentricity = 1/2.
24
DIRECTRIX
FOCUS
ELLIPSE
3
4
5
612
10
8
63
4
510
8
6
5
For each point on this hyperbola,Distance to Focus Point = Twice Distance to Directrix Line
Eccentricity = 2.
12
FO
CU
SP
OIN
T
HY
PE
RB
OL
A
DIR
EC
TR
IX L
INE
23
48
64
23
48
64
6
PARABOLAELLIPSE
CIRCLE HYPE
RBOL
A
Conic sections come from cutting a cone with a plane.The circle, ellipse, parabola and hyperbola
are all conic sections. 7
Conic Section means “Cut Cone”.A flashlight beam is a cone and the floor is a plane that cuts it.The circle, ellipse, parabola, and hyperbola are all conic sections.
8
With a hyperbola the floor cuts both halves of the light cone.There are two lines the hyperbola gets closer and closer to but
never touches. These are called the hyperbola’s asymptotes.9
For each point on this ellipse,Distance to Focus 1 + Distance to Focus 2 = 8.
10
26
35
44
35
62
Fo
cus
1F
ocu
s 2
For each point on this hyperbola,Distance to Focus 1 - Distance to Focus 2 = 4.
11
54
3
562
Focus2
Focus1
17
89
54
3
67
89
2
Tack two ends of a string to a sheet of drawing board.Keeping the string taut, move the pencil. The path will be an ellipse with a tack at each focus.
Planets, asteroids and comets move about our sun on ellipse shaped orbits.The sun lies at one focus of the ellipse. This is Kepler’s First Law.
The point closest to the sun is called the perihelion, the farthest point is the aphelion.
12
A Hohmann orbit from Earth to Mars is tangent to (just touches) the Earth orbit and Mars orbit.The Hohmann perihelion is at 1 A.U., the aphelion is at about 1.5 A.U.The Earth moves around the Sun at about 30 kilometers a second.
Mars moves around the Sun at about 24 kilometers a second.At perihelion the space ship is moving about 3 kilometers/second faster than earth.At aphelion, the spaceship is moving about 2.5 kilometers/second slower than Mars.
Mercury.39 A.U.
Venus.72 A.U.
Earth1 A.U.
Mars1.52 A.U.
The average distance from Earth’s center to the Sun’s center is called anastronomical unit, or A.U. for short. Mercury’s average distance from the
sun is .39 A.U., Venus .72 A.U. and Mars’ average distance is 1.52 A.U.
13
0.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3 1.4 1.5 1.6 1.7
1
a
ea
focus
b
a
ea
focus
b
Parts of an Ellipsea = semi major axisb = semi minor axis
e = eccentricity(in the above ellipse e = .5 or one half.)
ea = distance from ellipse center to focusThe semi major axis of an ellipse is often denoted with
the letter a. The semi minor axis is usually called b.An ellipses’ eccentricity is often labeled e.
14
e = 0e = 0
e = .3
e = .5
e = .6
e = .9
e = 1
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
0 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.1 1.2 1.3
In all of these ellipses a = 1. That is the semi major axis is one unit long.The circle is a special ellipse of eccentricity zero.
As eccentricity gets closer to one, the foci move from the center to the edge.A line segment could be regarded as an ellipse of eccentricity 1.
15
Aphelion
Perihelion
1month
1month
1m
onth1
month
1m
onth
1 month
1 month
1 month
1 month
1 month
1 month
1 month1 month1 month
1 month
1 month
1 month
1 mon
th
1 mon
th
1m
onth
1m
onth
1m
onth
1m
onth
1month1mo
nth
Over 1 month the orbit sweeps a wedge. Some wedges are short and fat, others tall and skinny.But they all have the same area.
An orbiting body sweeps equal areas in equal times.This is Kepler’s Second Law.
16
Perihelion
Aphelion
The two rectangles and parallelogram pictured above all have the same area.As an object gets closer to the sun it goes faster, so its velocity vector gets bigger. The
position vector (r) and velocity vector (v) make two sides of a parallelogram. The area of theparallelogram stays the same. At perihelion and aphelion the parallelogram is a rectangle.
17
r
v
r
v
r
v
22 = 2 X 2 = 42 squared is 2 times 2 which is 4.
Another way to read it:2 to the second power equals 2 times 2 which equals 4.
Can you see why 2 to the second power is also called 2 squared?
41/2 = 24 to the half power is 2. Or: The square root of 4 is 2.
32 = 3 X 3 = 93 squared is 3 times 3 which is 9.
Another way to read it:3 to the second power equals 3 times 3 which equals 9.
91/2 = 39 to the half power is 3. Or: The square root of 9 is 3.
42 = 4 X 4 = 164 squared is 4 times 4 which is 16.
161/2 = 416 to the half power is 4.
Or:The square root of 16 is 4.
Squares and Square RootsThis may not seem related to conic sections and orbital mechanics.
However, we will use these concepts in Kepler’s Third Law.18
You can also square fractions.
.52 = .5 X .5 = .25Marking off a square unit into
.1 x .1 squares results in 100 tiny squares.1/100 = .01
See how .01 looks like a penny?.5 X .5 has 25 tiny squares in it, so we write it as .25.
Half of a half is a quarter.1/2 X 1/2 = 1/4.
.251/2 = .5.25 to the half power is 5.
Or: The square root of .25 is .5.
.1
.2
.3
.4
.5
.6
.7
.8
.91.0
.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
.1
.2
.3
.4
.5
.6
.7
.8
.91.0
.1 .2 .3 .4 .5 .6 .7 .8 .91.0
1.11.2
.1.1 1.2
1.22 = 1.2 X 1.2 = 1.441.441/2 = 1.21.44 to the half power is 1.2.
Or: The square root of 1.44 is 1.2.
.2 X 1.0 = .20 .2 X .2 = .04
1.0 X 1.0 = 1.00
1.0 X .2 = .20
How do you multiplyDecimal fractions?
We can break the 1.2 X 1.2 square intotwo rectangles and two squaresThere are two skinny rectangles
Each row of 10 pennies is like a dime.Two dimes is .20
The big square is ten dimes or 1.00The little square is 4 pennies or .04
1.00 + .20 + .20 + .04 = 1.44.
1.2X 1.2
2 41 2 01.4 4
TwoDigits
past thedecimalpoint
Two Digitspast the
decimal point
Or you can multiply1.2 x 1.2 the same wayyou’d multiply 12 x 12.
However, count digits pastthe decimal point above the
line. There’s the samenumber of digits past thedecimal point in the answer.
19
Fractions as decimalsNot all quantities are in neat numbers like 1, 2, or 3.
Sometimes we need numbers more exact.
23 =2 X 2 X 2 =
2 X 4 =8
2 to the third power is 8.or:
2 cubed is 8.
81/3 = 28 to the one third power is 2.Or: The cube root of 8 is 2.
33 =3 X 3 X 3 =
3 X 9 =27
3 to the third power is 27.or:
3 cubed is 27.
271/3 = 327 to the one third power is 3.
The cube root of 27 is 3.
Cubes and Cube RootsThese are concepts also used in Kepler’s Third Law.
20
01
24
56
78
910
1112
1314
1516
173
The number of astronomical units of the semi-major axisraised to the 3/2 power gives the number of years a body
takes to orbit the sun. This comes from Kepler’s Third Law.
a = 16 A.U.This object
takes 64 yearsto orbit the sun.
163/2 =161/2 X 161/2 X 161/2 =
4 X 4 X 4 =4 X 16 = 64
a = 9 A.U.This object
takes 27 yearsto orbit the sun.
93/2 =91/2 X 91/2 X 91/2 =
3 X 3 X 3 =3 X 9 = 27
a = 4 A.U. This takes8 years to orbit the sun.
43/2 =41/2 X 41/2 X 41/2 =
2 X 2 X 2 = 8
a = 1 A.U. Earth takes1 year to orbit the sun.13/2 = 11/2 X 11/2 X 11/2 =
1 X 1 X 1 = 1
Saturn orbitsemi major axis
is about 9.54 A.U.9.543/2 = about 29.5.
Saturn’s orbital periodis about 29.5 years.
Jupiter orbitsemi major axisis about 5.2 A.U.
5.23/2 = about 11.9.Jupiter’s period
is about 11.9 years.
Mars orbitsemi major axis
is about 1.52 A.U.1.523/2 = about 1.9
Mars’ periodis about 1.9 years.
21
PYTHAGOREAN
THEOREM
22
Given a right triangle with legs a and b,and hypotenuse c,
a2 + b2 = c2
Both squares have side lengths (a+b)So the both have the same area: (a+b)2
+ = + =
=
23
12
513
62 + 82 = 102
36 + 64 = 100100 = 100
6
8
10
3
4
532 + 42 = 52
+ ==
+ = + =
=
8
15
17
ea ea
a ab
Both these arethe same length
a a
2a
24
Snip off the shorter string segment and put it on the other side and you’ll seethe string length is 2a, the length of the ellipse’s major axis.
b and ea are legsof a right trianglewith hypotenuse a.
(ea)2 + b2 = a2
b2 = a2 - (ea)2
b2 = (1 - e2)a2
b = (1 - e2)1/2a
25
A floating ball head is wearing a dunce cap/mosquito net. Where the ocean meets themosquito net is an ellipse. Where the ball head touches the water is a focus. Where the fishkisses the air is a focus. The ball head’s hat brim is a directrix plane as is the fish’s belt
plane. Where the directrix planes meet the ocean surface are two lines called directrix lines.
Each radius of a circle has length r.A line tangent to the circle is at rightangles to the radius it touches.by the Pythagorean theorem:e2 + r2 = f2 e2 = f2 - r2
g2 + r2 = f2 g2 = f2 - r2
e = gTwo such line segmentson tangent lines whoseend points meetare equal.
These lines tangent to a spheremeet at a point. The lines arecalled elements of a cone.
The bold line segmentsare all equal. The reasoningis similar to that inthe above panel: Eachline segment is a leg ofa right triangle, the otherleg being a radius of thesphere. All the righttriangles share thesame hypotenuse.
26
r r
fe g
27
If a = b and c = d, then a - c = b - d.It can be seen that each rib of the above lamp shade
is a line segment equal to each other rib.
- =
Each of these lamp shade ribline segments are equal
The elements of these conesare tangent to two spheres.The long line segments are
equal to each other.The short line segments are
equal to each other.
28
Dandelin spheres show that two descriptions ofthe ellipse do indeed describe the same thing.
r 1
r 2
L1
L2
This ellipse comes froma plane cutting a cone.
The plane cutting this cone is tangentto both Dandelin spheres.
Any line in this plane touching asphere is tangent to that sphere.
Because they’retwo meeting
tangent line segments,r1 = L1
andr2 = L2
r1 + r2 = L1 + L2
L1 + L2 is a Lampshade rib.
All the lampshade ribsare the same length.
r1 + r2 sum tothe same numberfor all points on
the ellipse.
Directrix
Focus
Distance
to Focus
Distance
to Directrix
These two
tangent lin
e
segments
are equal
Distance to Directr
ix
Distance to Focus
Drop a line segment straight down from the directrix plane to a point on the ellipse.The cone element line segment to the point is the same length as the point’s distance to the focus. All cone
elements meet the directrix plane at angle . The cutting plane meets the directrix plane at angle .The line straight down from the directrix is a fold in a triangle having angles and .
All these triangles are similar, having the same proportions.Since distance to focus and distance to directrix are always sides of similar triangles,
the ratio of these two lengths remain constant.Dandelin Spheres show that the focus and directrix definition of an ellipse describes the same thing.
29
30
We have looked at conics in terms of distance from a point and a line, and from two points.Now we will look at conics in terms of distance from two lines.
The vertical line we call the y axis, the horizontal line we call the x axis.Above is a picture of a parabola. Can you see a pattern between the
horizontal distances from the y axis and the vertical distances from the x axis?
x axis
y ax
is
y
x
(3, 9)
(2, 4)
(1, 1)
(0, 0)
(-1, 1)
(-2, 4)
(-3, 9)
y = x2
31
Above is the more usual way of showing a parabola on a Cartesian grid.When (x, y) coordinates are given, the first gives horizontal distancefrom the y axis, the second coordinate gives vertical distance from
the x axis. Going to the left or going down is given a minus sign.
y
x
(3, 4)
(0, 0)
x2 + y2 = 52
(4, 3)
(-3, 4)
(-4, 3)
(3, -4)
(4, -3)
(-3, -4)
(-4, -3)
(0, 5)
(0, -5)
(5, 0)(-5, 0) 345
32
The vertical and horizontal distance can be seen as legs of a right triangle. Thenthe distance from the origin (0, 0) to a point is the hypotenuse of this right triangle.
All these points are 5 units away from the origin.x2 + y2 = 52 describes a circle with radius 5.
r
r
r
r
rSlice a circle intosix wedges and re-arrange.
You have a shape that’sa bit more than aparallelogram withsides 3r by r.
Slice the circle intofiner wedges andre-arrange.The finer the wedges,the closer the circle is to arectangle having sides r and r. is a little more than 3. It’s about 3.14
r
3r
is a number a little more than 3, about 3.14. It’s spelled “pi” and pronounced “pie”,like delicious apple pie.
The area of a circle is r x r which is r2.For example a circle of radius 10 has an area of about 3.14 x 102, which is 314.
33
y
x
(3, 3.2)
(0, 0)
(4, 2.4)
(-3, 3.2)
(-4, 2.4)
(3, -3.2)
(4, -2.4)
(-3, -3.2)
(-4, -2.4)
(0, 4)
(0, -4)
(5, 0)(-5, 0) 34 5
x2 + ( y)2 = 5254
On page 24 we showed how the line from a focus to the top ofthe semi-minor axis (b) is the same length as the semi-major axis (a).
x2/a2 + y2/b2 = 1 describes an ellipsewith semi-major axis a and semi minor axis b.
x2 + ( y)2 = 5254
( )2(x2 + ( y)2) = ( )2(52)54
x2/52 + y2/42 = 1
15
15
34
x2/a2 + y2/b2 = 1
For this ellipsea = 5 and b = 4.
Can you figure out what e is?
35
(1 - e2) 1/2 2a
2a
Start with a circle having radius a.Vertically scale by b/a and you have an ellipsewith semi-major axis a and semi-minor axis b.
Each strip of the circle still has the same widthbut it’s height is shrunk by a factor of b/a.
Remember from page 33 that this circle has area a2.So the ellipse has area b/a x a2.
This is ab.Remember from page 24 that b = (1 - e2)1/2 a.
Area of an ellipse = (1 - e2)1/2 a2
V i n f
V i n f
V i n f
V i n f
V i n f
36
Remember on page 9 how a hyperbola getscloser and closer to the asymptotes?
As an object falls towards Earth, it movesfaster and faster. At the closest point tothe Earth, the perigee, it’s moving at topspeed. As it moves away, Earth’sgravity pulls it, slowing it down.As the hyperbola gets closerto the asymptote, the speedgets closer and closer toV infinity, the speed theobject would have atan infinite distancefrom Earth.
After a few millionkilometers from the Earth,
it is moving so close toV infinity, the difference
is negligible.
V infinity isalso called the
hyperbolic excess speed.
A ship leaves low Earthorbit entering ahyperbolic trajectorywith regard to the Earth.
A ship leaves low Earthorbit entering ahyperbolic trajectorywith regard to the Earth.
Zooming out we cansee that the hyperbolicpath approaches astraight line.
A ship leaves low Earthorbit entering ahyperbolic trajectorywith regard to the Earth.
37
Zooming out we cansee that the hyperbolicpath approaches astraight line.
We zoom outsome more.
What is this?The so called“straight lines”are starting togently curve.
What’s going onhere?
We’re entering ascale where the tinyEarth’s influence isbarely visible, butwe can start to seethe effects of themuch larger sun.
We zoom out even more and switch reference frames.No longer is the Earth our center. Thehyperbola with regard to Earthbecomes a tiny germ sittingon a much larger curve:The sun centeredEarth to Mars Hohmannellipse we sawon page 13.The 3 kilometers/secdifference betweenthe ellipse’sperihelion velocityaround the sun andEarth’s circularorbit velocityaround the Sunis the Vinfinityfor thehyperbolawith regardto Earth.
3 km/sec
If we zoom inwith a microscope
and switch referenceframes to Mars centered,
we’d see anothertiny hyperbola with
regard to Mars
VelocityCircularOrbit
38
VelocityParabolic
Orbit(also known as
EscapeVelocity)
VelocityHyperbolic
Orbit
V h y p e r b o l a
Vescape
V c i rc l e
V inf
init
y
Vcircle
V cir
cle
V esc
ape
V hyp
erbo
la
VelocityParabolic
Orbit(also known as
EscapeVelocity)
The further from a planet,the slower a circular orbit
At a given altitude,Vesc = 21/2 x Vcirc .The square root of 2is about 1.414.For a right isoscelestriangle, the hypotenuseis 21/2 x the length of eachof the two equal legs.
Likewise,Vhyp
2 = Vesc2 + Vinf
2.
Remember thePythagorean Theoremon pages 22 and 23?
Using the PythagoreanTheorem and the memorydevice to the right, it’s nothard to remember therelationships betweenVcirc, Vesc, Vhyp, and Vinf.
Parts of a Hyperbolaa = semi major axis = turning anglee = eccentricity
(in the above hyperbola e = 2.)
ea = distance from ellipse center to focus
-ea
-a
focus
-ea
-a
focus
The semi major axis of a hyperbola is often denotedwith the letter a. This is a negative number.A hyperbola’s eccentricity is often labeled e.
39
On page 28 was this equation: Vhyp2 = Vesc
2 + Vinf2.
Substituting for Vesc and Vinf:
Vhyp2 = ((2 x / r)1/2)2 + (( / -a)1/2)2.
Vhyp2 = (2 x / r) + ( / -a).
Vhyp2 = (2/r - 1/a).
Vhyp = ( (2/r - 1/a))1/2.
So if we know r and a, we can find the speed.This is called the vis-viva equation.V = ( (2/r - 1/a))1/2 also works for ellipses.
Since gravity gets weakerwith more distance, ittakes less speed to escape.
Escape velocity is(2 x / r)1/2.
V infinity is ( / -a)1/2.
G x mass is often called , pronounced “Moo”
Each speck of matter pullsother specks. You can think
of gravity as each specksending out tractor beams
More specks, more “tractor beams”.The more mass, the stronger the pull.
A body’s pull is G x mass.G is always the same, Mass is the amount of matter in a body.
40
r = 1
r = 2
r = 3
At distance r = 1,there’s 1 tractor beamper square unit.Double the distance,there’s 1 beam per4 square unitsTriple the distance,1 beam per 9 square units.Gravity’s pull gets weaker with distance.Acceleration from gravity is / r2.1 / r2 is called the inverse square. Gravity falls withthe inverse square or r.