Transcript PowerPoint

The Mathematics of Star
Trek
Lecture 2: Newton’s Three
Laws of Motion
Topics
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Functions
Limits
Two Famous Problems
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The Tangent Line Problem
Instantaneous Rates of Change
The Derivative
Velocity and Acceleration
Force
Newton’s Laws of Motion
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Functions
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What is a function?
Here is an informal definition:
A function is a procedure for assigning a
unique output to any acceptable input.
Functions can be described in many
ways!
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Example 1 (Some Functions)
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(a) Explicit algebraic formula
f(x) = 4x-5
(linear function)
g(x) = x2
(quadratic function)
r(x) = (x2+5x+6)/(x+2) (rational function)
p(x) = ex
(exponential function)
Functions f and g given above are also
called polynomials.
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Example 1 (cont.)
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(b) Graphical representation, such as the
following graph for the function y = x3+x.
y x3 x
10
5
0
-5
-10
-2
-1
0
1
2
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Example 1 (cont.)
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(c) Description or procedure
Assign to each Constitution Class starship in
the Federation an identifying number.
USS Enterprise is assigned NCC-1701
USS Excalibur is assigned NCC-26517
USS Defiant is assigned NCC-1764
USS Constitution is assigned NCC-1700
etc.
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Example 1 (cont.)
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(d) Table of values or data
In the Star Trek: The Original Series (TOS)
episode The Trouble With Tribbles, a furry little
animal called a tribble is brought on board the
USS Enterprise.
Three days later, the ship is overrun with
tribbles, which reproduce rapidly.
The following table gives the number of tribbles
on board the USS Enterprise, starting with one
tribble.
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Example 1 (cont.)
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(d) Tables of values or data (cont.)
Hours
Number of
Tribbles
0
1
12
11
24
121
36
1331
48
14641
60
456000
72
1771561
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Example 1 (cont.)
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(d) Tables of values or data (cont.)
The data in the table above describes a
function where the input is hours after
the first tribble is brought on board and
the output is the number of tribbles.
A natural question to ask is: When will
the USS Enterprise be overrun?
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Limits
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A concept related to function is the idea of a
limit.
The limit was invented to answer the question:
What happens to function values as input
values get closer and closer, but not equal to, a
certain fixed value?
If f(x) becomes arbitrarily close to a single
number L as x approaches (but is never equal
to) c, then we say the limit of f(x) as x
approaches c is L and write limx->c f(x) = L.
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Example 2 (Some limits)
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For each function given below, guess the
limit!
(a) limx->3 4x-5
(b) limx->-2 (x2+5x+6)/(x+2)
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Example 2 (cont.)
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(a) Make a table of values of the
function f(x) = 4x-5 for x-values near, but
not equal to, x = 3.
x
2
2.5
2.9
2.99
2.999
f(x)
3
5
6.6
6.96
6.996
x
4
3.5
3.1
3.01
3.001
f(x)
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9
7.4
7.04
7.004
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Example 2 (cont.)
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(b) Make a table of values of the function
r(x) = (x2+5x+6)/(x+2) for x-values near,
but not equal to, x = -2.
x
-3
-2.5
-2.1
-2.01
-2.001
r(x)
0
0.5
0.9
0.99
0.999
x
-1
-1.5
-1.9
-1.99
-1.999
r(x)
2
1.5
1.1
1.01
1.001
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Example 2 (cont.)
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(a) From the first table, it looks like:
limx->3 f(x) = limx->3 4x-5 = 7.
(b) From the second table it looks like:
limx->-2 r(x) = limx->-2 (x2+5x+6)/(x+2) = 1.
Notice that for the first function, we can put in x
= 3, but the second function is not defined at x
= -2!
This is one reason why limits were invented!!
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Two Famous Problems
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We now look at two famous
mathematical problems that lead to the
same idea!
The first problem deals with finding a line
tangent to a curve.
The second problem deals with find the
instantaneous rate of change of a
function.
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The Tangent Line Problem
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Studied by
Archimedes of
Syracuse (287-212
B.C)
In order to formulate
this problem, we
need to recall the
idea of slope.
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The Tangent Line Problem
(cont.)
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The slope of a line is the line’s rise/run.
Mathematically, we write: m = y/x.
For example, two points on the line y =
4x-5 are (x1,y1) = (0,-5) and (x2,y2) =
(3,7).
Therefore, the slope of this line is:
m = y/x = (y2-y1)/(x2-x1), i.e.
m = (7- -5)/(3 - 0) = 12/3 = 4.
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The Tangent Line Problem
(cont.)
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Graph of the line y = 4x - 5
y 4x 5
10
5
0
-5
-1
0
1
2
3
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The Tangent Line Problem
(cont.)
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Given the graph of a
function, y = f(x), the
tangent line at a point
P(a,f(a)) on the graph is
the line that best
approximates the
function at that point.
For example, the green
line is tangent to the
curve y = x3-x+4 at the
point P(1,-2).
y x3 x 4
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4
2
0
x
P
-2
-4
0
0.5
1
1.5
2
2.5
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The Tangent Line Problem
(cont.)
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The Tangent Line Problem is to find an
equation for the tangent line to the graph
of a function y = f(x) at the point
P(a,f(a)).
We’ll illustrate this problem with the
function:
y = f(x) = x3+x-4 at the point P(1,-2).
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The Tangent Line Problem
(cont.)
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To find the equation of a line, we need to
know two things:
A point on the tangent line,
The slope of the tangent line.
A point on the tangent line is P(1,-2).
To find the slope of the tangent line, we’ll
use the idea of secant lines.
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The Tangent Line Problem
(cont.)
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The slope of the red secant line
through the points P(a,f(a)) and
Q(a+h,f(a+h)) is given by:
mPQ = f/x = (f(a+h)-f(a))/h
To find the slope of the green
tangent line, let h->0, i.e.
mtan = limh->0 (f(a+h)-f(a))/h
In this case, we find that the
slope of the tangent line to the
graph of y = f(x) at P is mtan = 4.
Using the point-slope form of a
line, an equation for the tangent
line is:
y - (-2) = 4(x-1), or y = 4x - 6.
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Q
4
2
0
x
P
-2
-4
0
0.5
1
1.5
2
2.5
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The Rate of Change Problem
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This problem was
studied in various forms
by:
Johannes Kepler (15711630)
Galileo Galilei (15641642)
Isaac Newton (16431727)
Gottfried Leibnitz (16461716)
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The Rate of Change Problem
(cont.)
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Here’s an example to motivate this problem:
While flying the shuttlecraft back to the
Enterprise from Deneb II, Scotty realizes that
the shuttlecraft’s speedometer is broken.
Fortunately, the shuttlecraft’s odometer still
works.
How can Scotty measure his velocity?
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The Rate of Change Problem
(cont.)
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Let s = f(t) = distance in kilometers the
shuttlecraft is from Deneb II at time t ≥0
seconds.
Here s is the shuttlecraft’s odometer
reading.
Assume Scotty has zeroed out the
odometer at time t = 0 seconds.
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The Rate of Change Problem
(cont.)
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The average velocity of the shuttlecraft
between times a and a+h is:
vave = f/t = (f(a+h)-f(a))/h
The velocity (or instantaneous velocity)
of the shuttlecraft is the quantity we get
as h -> 0 in the expression for average
velocity, i.e.
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The Rate of Change Problem
(cont.)
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The velocity at time t = a is:
v = limh->0 (f(a+h)-f(a))/h.
The velocity is the instantaneous rate of
change of the position function f with
respect to t at time t = a.
Thus, Scotty can estimate his velocity at
time t = a by computing average
velocities over short periods of time h.
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The Rate of Change Problem
(cont.)
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This idea can be generalized to other
functions.
If y = f(x), the average rate of change of f
with respect to x between x = a and x =
a+h is f/x = (f(a+h)-f(a))/h.
The instantaneous rate of change of f
with respect to x at the instant x = a is
given by limh->0 (f(a+h)-f(a))/h.
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The Derivative
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Notice that the two problems we just
looked at lead to the same result - a limit
of the form limh->0 (f(a+h)-f(a))/h.
Thus, finding the slope of a tangent line
is exactly the same thing as finding an
instantaneous rate of change!
We call this common quantity found by a
limit the derivative of f at x = a!
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The Derivative (cont.)
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The derivative of a function f(x) at the
point x = a, denoted by f’(a), is found by
computing the limit:
f’(a) = limh->0 (f(a+h)-f(a))/h, provided this
limit exists!
Note: we call (f(a+h)-f(a))/h a difference
quotient.
Thus, for f(x) = x3+x-4, f’(1) = 4.
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The Derivative (cont.)
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Since at each x = a, we get a slope, f’(a), f’ is
really a function of x!
Thus, we can make up a new function!
Given a function f, the derivative of f, denoted
f’, is the function defined by:
f’(x) = limh->0 (f(x+h)-f(x))/h, provided this limit
exists!
Other notation for f’(x) includes that due to
Leibnitz: dy/dx or d/dx[f(x)] .
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Ways to Find a Derivative
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Mathematicians have figured out “shortcuts” to find
derivatives of functions.
If f(x) = k, where k is a constant, then f’(x) = 0.
If g(x) = k f(x), where k is a constant and f’ exists,
then g’(x) = k f’(x).
If h(x) = f(x) + g(x) and f’ and g’ exist, then
h’(x)
= f’(x) + g’(x).
If f(x) = xn, where n is a rational number, then
f’(x)
= n xn-1.
If f(x) = ek x, where k is a constant, then f’(x) = k ek x.
For example, the derivative of f(x) = x3+x-4 is
f’(x) = 3x2+1. Notice that f’(1) = 3(1)2+1=4.
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Velocity and Acceleration as
Functions
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If an object is in motion, then we can talk about
its velocity, which is the rate of change of the
object’s position as a function of time.
Thus, at every moment in time, a moving
object has a velocity, so we can think of the
object’s velocity as a function of time!
This in turn implies that we can look at the rate
of change of an object’s velocity function via
the derivative of the velocity function.
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Velocity and Acceleration as
Functions (cont.)
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The acceleration of an object is the
instantaneous rate of change of it’s velocity
with respect to time.
Thus, if s(t) gives an object’s position, then
v(t) = s’(t) gives the object’s velocity and
a(t)
= v’(t) gives the object’s acceleration.
We call the acceleration the second derivative
of the position function.
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Force
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Force is one of the foundational
concepts of physics.
A force may be thought of as any
influence which tends to change the
motion of an object.
Physically, force manifests itself when
there is an acceleration.
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Force (cont.)
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For example, if we are on board the Enterprise when it
accelerates forward, we will feel a force in the opposite
direction that pushes us back into our chair.
There are four fundamental forces in the universe, the
gravity force, the nuclear weak force, the electromagnetic
force, and the nuclear strong force in ascending order of
strength.
Isaac Newton wrote down three laws that describe how
force, acceleration, and motion are related.
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Newton’s First Law of Motion
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Newton’s first law is based on
observations of Galileo.
Newton’s First Law: An object will
remain at rest or in uniform motion in a
straight line unless acted upon by an
external force.
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Newton’s First Law of Motion
(cont.)
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The property of objects that makes them “tend”
to obey Newton’s first law is called inertia.
Inertia is resistance to changes in motion.
The amount of inertia an object has is
measured by its mass.
For example, a starship will have a lot more
mass than a shuttlecraft.
It will take a lot more force to change the
motion of a starship!
A common unit for mass is the kilogram.
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Newton’s Second Law of Motion
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Newton’s second law relates force, mass and
acceleration:
Newton’s Second Law: The net external force
on an object is equal to its mass times
acceleration, i.e. F = ma.
The weight w of an object is the force of gravity
on the object, so from Newton’s second law,
w = mg, where g is the acceleration of gravity.
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Newton’s Third Law of Motion
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Newton’s Third Law: All forces in the
universe occur in equal but oppositely
directed pairs. There are no isolated
forces; for every external force that acts
on an object there is a force of equal
magnitude but opposite direction which
acts back on the object which exerted
that external force.
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References
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Calculus: Early Transcendentals (5th ed) by
James Stewart
Hyper Physics: http://hyperphysics.phyastr.gsu.edu/hbase/hph.html
Memory Alpha Star Trek Reference:
http://memory-alpha.org/en/wiki/Main_Page
The Cartoon Guide to Physics by Larry Gonick
and Art Huffman
St. Andrews' University History of
Mathematics: http://www-groups.dcs.stand.ac.uk/~history/index.html
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