The Mathematics of Star Trek

1
The Mathematics of Star Trek

• Lecture 2 Newtons Three Laws of Motion

2
Topics

• Functions
• Limits
• Two Famous Problems
• The Tangent Line Problem
• Instantaneous Rates of Change
• The Derivative
• Velocity and Acceleration
• Force
• Newtons Laws of Motion

3
Functions

• 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)

• (a) Explicit algebraic formula
• f(x) 4x-5 (linear function)
• g(x) x2 (quadratic
  function)
• r(x) (x25x6)/(x2) (rational function)
• p(x) ex (exponential function)
• Functions f and g given above are also called
  polynomials.

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Example 1 (cont.)

• (b) Graphical representation, such as the
  following graph for the function y x3x.

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Example 1 (cont.)

• (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.)

• (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.)

• (d) Tables of values or data (cont.)

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Example 1 (cont.)

• (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

• 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-gtc f(x) L.

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Example 2 (Some limits)

• For each function given below, guess the limit!
• (a) limx-gt3 4x-5
• (b) limx-gt-2 (x25x6)/(x2)

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Example 2 (cont.)

• (a) Make a table of values of the function f(x)
  4x-5 for x-values near, but not equal to, x 3.

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Example 2 (cont.)

• (b) Make a table of values of the function r(x)
  (x25x6)/(x2) for x-values near, but not equal
  to, x -2.

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Example 2 (cont.)

• (a) From the first table, it looks like
• limx-gt3 f(x) limx-gt3 4x-5 7.
• (b) From the second table it looks like
• limx-gt-2 r(x) limx-gt-2 (x25x6)/(x2) 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

• 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

• 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.)

• The slope of a line is the lines 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.)

• Graph of the line y 4x - 5

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The Tangent Line Problem (cont.)

• 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-x4 at the point P(1,-2).

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The Tangent Line Problem (cont.)

• 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)).
• Well illustrate this problem with the function
• y f(x) x3x-4 at the point P(1,-2).

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The Tangent Line Problem (cont.)

• 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, well use
  the idea of secant lines.

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The Tangent Line Problem (cont.)

• The slope of the red secant line through the
  points P(a,f(a)) and Q(ah,f(ah)) is given by
• mPQ ?f/?x (f(ah)-f(a))/h
• To find the slope of the green tangent line, let
  h-gt0, i.e.
• mtan limh-gt0 (f(ah)-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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The Rate of Change Problem

• This problem was studied in various forms by
• Johannes Kepler (1571-1630)
• Galileo Galilei (1564-1642)
• Isaac Newton (1643-1727)
• Gottfried Leibnitz (1646-1716)

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The Rate of Change Problem (cont.)

• Heres an example to motivate this problem
• While flying the shuttlecraft back to the
  Enterprise from Deneb II, Scotty realizes that
  the shuttlecrafts speedometer is broken.
• Fortunately, the shuttlecrafts odometer still
  works.
• How can Scotty measure his velocity?

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The Rate of Change Problem (cont.)

• Let s f(t) distance in kilometers the
  shuttlecraft is from Deneb II at time t 0
  seconds.
• Here s is the shuttlecrafts odometer reading.
• Assume Scotty has zeroed out the odometer at time
  t 0 seconds.

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The Rate of Change Problem (cont.)

• The average velocity of the shuttlecraft between
  times a and ah is
• vave ?f/?t (f(ah)-f(a))/h
• The velocity (or instantaneous velocity) of the
  shuttlecraft is the quantity we get as h -gt 0 in
  the expression for average velocity, i.e.

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The Rate of Change Problem (cont.)

• The velocity at time t a is
• v limh-gt0 (f(ah)-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.)

• 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 ah is ?f/?x
  (f(ah)-f(a))/h.
• The instantaneous rate of change of f with
  respect to x at the instant x a is given by
  limh-gt0 (f(ah)-f(a))/h.

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The Derivative

• Notice that the two problems we just looked at
  lead to the same result - a limit of the form
  limh-gt0 (f(ah)-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.)

• 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-gt0 (f(ah)-f(a))/h, provided this
  limit exists!
• Note we call (f(ah)-f(a))/h a difference
  quotient.
• Thus, for f(x) x3x-4, f(1) 4.

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The Derivative (cont.)

• 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-gt0 (f(xh)-f(x))/h, provided this
  limit exists!
• Other notation for f(x) includes that due to
  Leibnitz dy/dx or d/dxf(x) .

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Ways to Find a Derivative

• 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) x3x-4 is
  f(x) 3x21. Notice that f(1)
  3(1)214.

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Velocity and Acceleration as Functions

• If an object is in motion, then we can talk about
  its velocity, which is the rate of change of the
  objects position as a function of time.
• Thus, at every moment in time, a moving object
  has a velocity, so we can think of the objects
  velocity as a function of time!
• This in turn implies that we can look at the rate
  of change of an objects velocity function via
  the derivative of the velocity function.

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Velocity and Acceleration as Functions (cont.)

• The acceleration of an object is the
  instantaneous rate of change of its velocity
  with respect to time.
• Thus, if s(t) gives an objects position, then
  v(t) s(t) gives the objects velocity and
  a(t) v(t) gives the objects acceleration.
• We call the acceleration the second derivative of
  the position function.

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Force

• 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.)

• 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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Newtons First Law of Motion

• Newtons first law is based on observations of
  Galileo.
• Newtons 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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Newtons First Law of Motion (cont.)

• The property of objects that makes them tend to
  obey Newtons 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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Newtons Second Law of Motion

• Newtons second law relates force, mass and
  acceleration
• Newtons 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 Newtons second law, w
  mg, where g is the acceleration of gravity.

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Newtons Third Law of Motion

• Newtons 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

• Calculus Early Transcendentals (5th ed) by
  James Stewart
• Hyper Physics http//hyperphysics.phy-astr.gsu.e
  du/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.st-and.ac.uk/history/index.
  html