The Mathematics of Star Trek

1
The Mathematics of Star Trek

• Lecture 4 Motion in Space and Keplers Laws

2
Topics

• Vectors
• Vector-Valued Functions
• Motion in Space
• Keplers Laws

3
Vectors

• A vector is a quantity with a magnitude and a
  direction.
• Examples of vectors
• Force
• Displacement
• Velocity
• Acceleration
• A scalar is a real number.
• Examples of scalars
• Mass
• Length
• Speed
• Time

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Vectors (cont.)

• We can represent a vector in two ways
• Graphically, we can represent vectors with an
  arrow.
• The length of the arrow gives the magnitude of
  the vector.
• The arrow points in the direction of the vector.
• Here, vectors u and v have the same length and
  magnitude, hence are equal.

u
v
w
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Vectors (cont.)

• We can also represent a vector algebraically.
• The vector v lt2,3gt is graphed to the right, in
  standard position.
• Vector v ltv1,v2gt has components v1 2 and
  v2 3.
• Using the Pythageorean Theorem, we find that
  vector v has magnitude v (v12v22)1/2
  (2232)1/2 131/2.

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Vectors (cont.)

• We can add two vectors to get a new vector!
• Here is how to add the vectors u and v
  graphically.
• Move the vector v so its tail corresponds with
  the head of vector u.
• Then draw a vector with tail at the tail of u and
  head at the head of v.

u
v
uv
v
u
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Vectors (cont.)

• To add two vectors algebraically, we use the
  following idea
• If ultu1,u2gt and vltv1,v2gt, then
  uvltu1v1,u2v2gt.
• For example, if ult2,3gt and vlt-1,4gt, then
• uv lt2(-1),34gt lt1,7gt.
• Do the graphical and algebraic adding methods
  agree?

uv
v
u
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Vectors (cont.)

• Another operation we can perform on vectors is
  scalar multiplication.
• Graphically, the vector kv is the vector of
  length k times the magnitude of v, pointing in
  the same (opposite) direction as v if kgt0 (klt0).
• Algebraically, given vltv1,v2gt,
• kvltkv1,kv2gt.
• For example, if k3 and vlt2,3gt,
• kv2lt2,3gtlt4,9gt.

v
kv
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Vector-Valued Functions

• A vector-valued function is a function whose
  inputs are real numbers and outputs are vectors!
• A vector-valued function has the form
• F(t) ltf(t),g(t)gt (plane vector),
• F(t) ltf(t),g(t),h(t)gt (space vector).
• We call f, g, and h, component functions of
  vector function F.
• The graph of a vector-valued function is the set
  of points traced out by the head of the vector
  r(t) as t varies.

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Vector-Valued Functions (cont.)

• Here is the graph of the vector-valued function
• F(t) lt2 cos t, sin tgt

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Vector-Valued Functions (cont.)

• Here is the graph of the vector-valued function
• G(t) ltt - 2 sin t, 2 - 2 cos tgt

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Vector-Valued Functions (cont.)

• Here is the graph of the vector-valued function
• H(t) ltcos t, sin t, t2gt
• Show Mathematica examples!

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Vector-Valued Functions (cont.)

• Just like the functions we saw before, we can
  find limits and derivatives of vector-valued
  functions!
• Given F(t) ltf(t),g(t)gt we define
• limt-gta F(t) lt limt-gta f(t), limt-gta g(t)gt,
  provided the component function limits all exist,
  and
• F(t) ltf(t), g(t)gt.
• Similar definitions hold for space vector
  functions!

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Vector-Valued Functions (cont.)

• Example Given the derivative rules
• d/dtsin t cos t and d/dtcos t -sin t,
• find the derivatives of the vector-valued
  functions
• F(t) lt2 cos t, sin tgt
• G(t) ltt - 2 sin t, 2 - 2 cos tgt
• H(t) ltcos t, sin t, t2gt
• Question If these vector functions describe an
  objects position, what might the derivative
  describe?

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Motion in Space

• If an objects position in space (or the plane)
  is described by the vector function r(t), then
  its velocity and acceleration are given by
• v(t) r(t),
• a(t) v(t) r(t).
• Graphically, the velocity and acceleration
  vectors are drawn with their tails at the head of
  position vector r(t), which is drawn in standard
  position.

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Motion in Space (cont.)

• For example, here is the graph of the
  vector-valued function G(t) ltt - 2 sin t, 2 - 2
  cos tgt along with its velocity vector (blue) and
  acceleration vector (green) for a fixed t-value!
• The length of the velocity vector corresponds to
  the objects speed.
• Notice that the acceleration vector is points to
  the inside of the curve!
• Show Mathematica examples!

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Motion in Space (cont.)

• We can also talk about integration of
  vector-valued functions!
• Given vector-valued function F(t) ltf(t),g(t)gt,
  we define the integral of F by
• ? F(t) dt lt ? f(t) dt, ? g(t) dtgt.
• For example, lets find ? F(t) dt, if
• F(t) ltt, 3t21gt.
• Solution ? F(t) dt lt1/2 t2, t3tgt ltC1,C2gt.

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Laws of Motion (Revisited)

• As we saw before, one application of integration
  is to find equations of motion for an object!
• The same ideas work for vector-valued functions!
• Example An object moves with constant
  acceleration a. Find the objects velocity,
  given an initial velocity of v0 at time t 0.
• Solution
• v(t) ? a dt a t C.
• v0 v(0) a (0) C,
• v0 C,
• v(t) a t v0.
• Homework Find the objects position, given an
  initial position of s0 at time t 0.

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Keplers Laws

• One application of vector-valued functions is to
  describe the motion of planets!
• After spending 20 years studying planetary data
  collected by the Danish astronomer Tycho Brahe
  (1546-1601), the German mathematician Johannes
  Kepler (1571-1630) formulated the following three
  laws

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Keplers Laws (cont.)

• 1. A planet revolves around the Sun in an
  elliptical orbit with the Sun at one focus. (Law
  of Orbits)
• 2. The line joining the Sun to a planet sweeps
  out equal areas in equal times. (Law of Areas)
• 3. The square of the period of revolution of a
  planet is proportional to the cube of the length
  of the semimajor axis of its orbit. (Law of
  Periods)

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Keplers Laws (cont.)

• Kepler formulated these laws because they fit the
  measured data, but could not see why they should
  be true or how they were related.
• In 1687, in his Principia Mathematica, Sir Isaac
  Newton was able to show that all three laws
  follow from vector formulations of his universal
  law of gravitation and second law.
• For a discussion of this derivation, see
  Stewarts Calculus - Early Transcendentals (5th
  ed.), pp. 880 - 881.

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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
• St. Andrews' University History of Mathematics
  http//www-groups.dcs.st-and.ac.uk/history/index.
  html