The Mathematics of Star Trek Lecture 2: Newton’s Three Laws of Motion

The Mathematics of Star Trek

Lecture 2: Newton’s Three Laws of Motion

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Topics

Functions Limits Two Famous Problems

• The Tangent Line Problem

• Instantaneous Rates of Change The Derivative Velocity and Acceleration Force Newton’s Laws of Motion

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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) = (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.)

(b) Graphical representation, such as the following graph for the function y = x3+x.

-2 -1 0 1 2-10

-5

0

5

10yx3x

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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.)

HoursNumber of

Tribbles

0 1

12 11

24 121

36 1331

48 14641

60 456000

72 1771561

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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->c f(x) = L.

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Example 2 (Some limits)

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.)

(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) 11 9 7.4 7.04 7.004

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Example 2 (cont.)

(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.)

(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

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 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.) Graph of the line y = 4x - 5

-1 0 1 2 3 4

-5

0

5

10

y4x5

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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).

0 0.5 1 1.5 2 2.5

-4

-2

0

2

4

6

x

yx3x4

P

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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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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.) 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.

0 0.5 1 1.5 2 2.5

-4

-2

0

2

4

6

xP

Q

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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.)

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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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 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 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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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

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.)

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.)

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 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

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.)

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

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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Newton’s First Law of Motion

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.) 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

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

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 Calculus: Early Transcendentals (5th ed) by

James Stewart Hyper Physics:

http://hyperphysics.phy-astr.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.st-and.ac.uk/~history/index.html

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The Mathematics of Star Trek Lecture 2: Newton’s Three Laws of Motion