The Mathematics of Star Trek

Lecture 4: Motion in Space and Kepler’s Laws

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Topics

Vectors Vector-Valued Functions Motion in Space Kepler’s Laws

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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

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Vectors (cont.) We can also represent a

vector algebraically. The vector v = <2,3> is

graphed to the right, in standard position.

Vector v = <v1,v2> has components v1 = 2 and v2 = 3.

Using the Pythageorean Theorem, we find that vector v has magnitude |v| = (v1

2+v22)1/2 =

(22+32)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.

uv

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Vectors (cont.) To add two vectors

algebraically, we use the following idea:

If u=<u1,u2> and v=<v1,v2>, then u+v=<u1+v1,u2+v2>.

For example, if u=<2,3> and v=<-1,4>, then

u+v = <2+(-1),3+4> = <1,7>.

Do the graphical and algebraic adding methods agree?

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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 k>0 (k<0).

Algebraically, given v=<v1,v2>, kv=<kv1,kv2>.

For example, if k=3 and v=<2,3>,kv=2<2,3>=<4,9>.

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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) = <f(t),g(t)> (plane vector), F(t) = <f(t),g(t),h(t)> (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) = <2 cos t, sin t>

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Vector-Valued Functions (cont.)

Here is the graph of the vector-valued function:G(t) = <t - 2 sin t, 2 - 2 cos t>

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Vector-Valued Functions (cont.)

Here is the graph of the vector-valued function:H(t) = <cos t, sin t, t2>

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) = <f(t),g(t)> we define:

limt->a F(t) = < limt->a f(t), limt->a g(t)>, provided the component function limits all exist, andF’(t) = <f’(t), g’(t)>.

Similar definitions hold for space vector functions!

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Vector-Valued Functions (cont.) Example: Given the derivative rules:

d/dt[sin t] = cos t and d/dt[cos t] = -sin t, find the derivatives of the vector-valued functions:• F(t) = <2 cos t, sin t>• G(t) = <t - 2 sin t, 2 - 2 cos t>• H(t) = <cos t, sin t, t2>

Question: If these vector functions describe an object’s position, what might the derivative describe?

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Motion in Space

If an object’s 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) = <t - 2 sin t, 2 - 2 cos t> along with its velocity vector (blue) and acceleration vector (green) for a fixed t-value!

The length of the velocity vector corresponds to the object’s 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) = <f(t),g(t)>, we define the integral of F by:

∫ F(t) dt = < ∫ f(t) dt, ∫ g(t) dt>. For example, let’s find ∫ F(t) dt, if

F(t) = <t, 3t2+1>. Solution: ∫ F(t) dt = <1/2 t2, t3+t> + <C1,C2>.

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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 object’s 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 object’s position, given an initial position of s0 at time t = 0.

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Kepler’s 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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Kepler’s 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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Kepler’s 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 Stewart’s 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.edu/hbase/hph.html

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