Newton's Derivvation of Orbital
Mottion
By Ca
alvin Li
Kepler'ss Laws
• K
Kepler
l observed
b
d th
three llaws of
of planetary
l
t
motion
ti
o Never actually proved them
m
• (1) Every planet orbits around
d the sun in an ellipse with the
Sun being one of the two focii
• (2) Area swept in a given time
e is always constant
• (3) The square of the orbital period
p
is directly proportional to
the cube of the semi-major
semi major axxis
Newton Derives Kepler's
K
First Law
• N
Newton
t ttookk on the
th task
t k off prroving
i allll off K
Kepler's
l ' Th
Three
Laws
• More importantly, Newton generalized Kepler's first law by
proving that a satellite in orbitt of a celestial being will move
in the shape of a conic sectio
on
General: Conic Sections
•
Circle is made by cutting a plane
parallel to base
•
Ellipse is made by cutting a plane
slightly not parallel to base
•
Parabola is made by cutting a plane
parallel to side of cone
•
Hyperbola is made by cutting a plane
perpendicular to base
Circles and Ellipses
Circles:
Ci
l
A shape
h
iin which
hi h allll th
the
points are equidistant to a focus
• A special case of an ellipse in
which both axis are equal lengths
(the radius)
Ellipses: A shape in which the sum
of the distance from one focus to a
point on the shape and the distance of
the other focus to the same point is
constant
Parabolas and Hyperbolas
Parabolas:
P
b l
A curved
d liline iin which
hi h th
the
distance between the focus and a
point on the curved line is equal to
the distance between that point to a
point on the straight line (directrix)
Hyperbolas: A curved lines in which
the differences of the two focis from all
points are always constant
Proof: Part 1
Proof: Part 2
Proof: Part 3
Proof: Part 4
Bibliog
graphy
• Professor
P f
Hafez
H f
http://www.math.utk.edu/
math utk edu/~fre
freire/teaching/fall2006/m142f06N
• http://www
ewtonKepler.pdf
• http://www.braeunig.us/space
e/orbmech.htm
• http://mathworld
http://mathworld.wolfram.com
wolfram com
m/images/epsm/images/eps
gif/ConicSection_1000.gif
• http://en.wikipedia.org/wiki/Co
onic_sections
• http://en.wikipedia.org/wiki/O
http://en wikipedia org/wiki/Orbit
Sir Isaacc Newton
By Ca
alvin Li
Basic Information
•
•
•
•
•
Birth Date: December 25, 1642
Birth Place: Woolsthrope
Woolsthrope, England
Large Contribution in his Study of Math and
Physics
President of Royal Society in London 17031727
Death Date: March 20, 1727
Contribution to Math: Calculus
• C
Created
t d calculus
l l to
t solve
l fifinit
itte
t differences
diff
and
d areas under
d
the curve (small delta x)
• Differentiation = Fluxion
• Integrate = Fluents
• Leibniz notation vs. Newton notation
n
• Fundamental Theorem of Calculus
Application of Calculus
• Newton's
N t ' method
th d tto solve
l roo
ots
t off a reall function
f
ti
• Infinite series
o Did not focus much on it
o Claimed to have known Ta
aylor series before discovery
Binomial Expan
nsion Theorem
• C
Created
t d generalized
li d th
theorem
m to
t expand
d the
th sum off two
t
terms raised to a given power
Pascal s Triangle a
and Combinations
• Related to Pascal's
Newton's Famo
ous Apple Story
• H
He once ttold
ld a story
t
off seeing
i g an apple
l ffallll off
ff off an apple
l
tree and thinking about the fo
orces causing the apple to fall
o Cartoons change story of a
apple falling on Newton's
Newton s head
• Newton questions how the la
aws of nature govern motion of
objects
• Led to the discovery of Newto
on's Four Laws of Motion
Newton's Firstt Law of Inertia
• "E
"Every b
body
d persists
i t iin itits sta
tate
t off being
b i att restt or off moving
i
uniformly straight forward, exxcept insofar as it is compelled
to change its state by force im
mpressed
mpressed"
o Only is true in an inertial re
eference frame
• Concept of inertia
Newton's Second
S
Law
• "Th
"The alteration
lt ti off motion
ti is
i ever proportional
ti
l tto th
the motive
ti
force impress'd; and is made in the direction of the right line
in which that force is impresss'd"
sd
• The sum of all the force vecto
ors is equal to the rate at which
momentum changes.
o Net Force = d(mv)/dt
• Simplied version of the law: Fnet = ma
• Concept of Impulse
o J = integral(F)dt
o J = F*t
Newton's Third Law
• "T
"To every action
ti th
there is
i alw
lways an equall and
d opposite
it
reaction: or the forces of two bodies on each other are
always equal and are directed in opposite directions
directions."
• Example: Given Mass on Floor
Newton's Law of Un
niversal Gravitation
• E
Every point
i t mass iis attracted
tt t d tto any other
th point
i t mass with
ith a
pulling force directly proportio
onal to the mass of both point
masses and inversely proporrtional to the square of the
distance between them
o F = G (m1*m2/r2)
o G is gravitational constant (=6.674 x 1011 N*m2/kg2)
• Helped Newton prove Keplerr's Laws and understand how
satellites orbit
Contribution to Light
L
and Optics
• IInvention
ti off shorter
h t telescope
t l
e using
i reflecting
fl ti mirrors
i
• Stated that light acts as a parrticle (massless bundle of
energy)
Bibliog
graphy
• Professor
P f
Hafez
H f
http://en wikipedia org/wiki/Isaac_newton
aac newton
• http://en.wikipedia.org/wiki/Is
• http://www.maths.tcd.ie/pub/H
HistMath/People/Newton/Rouse
Ball/RB_Newton.html
• http://ptri1.tripod.com/ptreal1
http://ptri1 tripod com/ptreal1r.gif
r gif
• http://amazingp
g
space.stsci.edu/resources/exxplorations/groundup/lesson/sco
pes/newton/graphics/tele_new
wton_small.jpg