Note: This lesson uses vectors, and some way of denoting them on the board
and in the notebook must be agreed on by the class. In this lesson plan, all vector quantities will be underlined.
Goals: The student will learn
- About the definition and purpose of vectors, in mathematics and physics.
- To use vector addition for representing the sum of two motions taking place simultaneously.
- To resolve vectors into components along the directions of given axes,
in two or three dimensions.
- To add two or more vectors, using components
- To resolve forces on an object that rests on an inclined plane.
Terms: Vector, vector addition, vector components, magnitude of a vector, vector components parallel and perpendicular to a given direction.
Stories and extras: None here; however section #22a on airplane flight has some interesting applications, which could follow this lesson.
Today we discuss vectors, mathematical objects which have not only a magnitude, a size, the way ordinary numbers have, but also a direction in which they point. They can be approached in different ways.
They can be viewed as a wider definition of numbers. Numbers can be defined in stages, each stage generalizing the previous one but covering a wider class, like circles within circles. (Illustrate on the board by a line on which numbers are marked, also write underlined terms in a table--each new one below the preceding ones.).
The earliest numbers were integers: 1,2,3,4 .... and so on, invented very early, for practical purposes--say, counting sheep as they came home, to make sure none is missing.
Then negative numbers: –1, –2, –3... --you owe me one, two, three sheep. Also zero, which was only regarded as a number fairly late.
Then fractions--1/2, 1/3, also 7/12 or 3/7 and so on; the Egyptians only knew the first kind, and would write the 3rd and 4th fractions as (1/2)+(1/12) and as (1/3)+(1/12)+(1/84). Also decimal fractions.
Then "irrational numbers" such as the square root of 2 which cannot be written as any fraction (there is a simple proof). All these together are known as real numbers.
What next?. Several ways exist of extending the concept of numbers to still wider classes--which along with real numbers, include additional quantities which can be manipulated.
Of course, we need give some for those additions. Ane real number can be viewed as the length of a lone. With wider definitions, such simple interpretations may no longer work.
For instance, we may include(complex numbers) which inlude i, the square root of (–1), and expressions such as a + bi, where a and b are real numbers. That is a direction in which we will not go today (which is why the term was written in parentheses). It may be noted in passing, however, that complex numbers have a close connection to vectors in 2 dimensions.
So instead, what will it be? All the above can be related to points along a line: integers are isolated points, fractions seem to fill the spaces between them quite densely, but they still leave enough space to squeeze in the irrationals.
Now, presumably, all the points on the line are covered. For each number we can put an arrow on the line, the distance from zero to that number--arrows to the right (say) for positive numbers, to the left for negative ones.
Vectors are mathematical objects that represent arrows in any direction--in the plane, even in 3 dimensions. It is a new level of "numbers", and that is one way of looking at them.
In algebra, we mark ordinary numbers ("scalars") with letters.
If we want to show a quantity is a vector, mark it with an arrow above, or an underline or (mainly in books) in bold face. In the web files of "Stargazers," unfortunately, bold face is used to highlight quantities, so this convention is not followed, and you will have to distinguish vectors from their context.
Mathematicians have invented all sort of strange generalizations of numbers. The ones of most interest are the ones with good applications.
Vectors allow us to represent velocities.
We fly an airplane and meanwhile the wind pushes it sideways--how are we progressing relative to the ground? Vectors help answer that.
Similarly, forces, accelerations, magnetc fields from several sources, all are added like vectors. Engineers who put up a bridge or a building and want to make sure all forces balance, etc., need vectors.
Enough talking about them--any examples?
The simplest kind is displacement (sketch on the board a map of the US and use it). You take a pencil and displace it from New York to Chicago, then from Chicago to Seattle. The final effect is the same as if we displaced the pencil from New York to Seattle.
The displacement from New York to Chicago is this arrow.
From Chicago to Seattle -- this arrow
From New York to Seattle --this arrow, and we say it is the vector sum of the other two arrows.
It may look like a strange way of adding--but that is also how you add velocities, and forces, and magnetic fields.
(now to the lesson)
Guiding questions and additional tidbits with suggested answers.
--What is the graphical method of adding two vectors?
Place the tail of the second at the head of the first--the sum is from the tail of the first to the head of the second
--Does it make any difference which of the two is added first and which second?
-- Why? (The teacher demonstrates on the board.
Say we add two vectors a and b.
a + b gives one triangle
b + a gives a mirror-image triangle.
Both triangles can be combined to a single parallelogram (show on the blackboard). In either case, the sum is the diagonal of the parallelogram--the same diagonal in both cases.
-- When do vectors add like numbers?
When they all are along the same line.
--But vectors along a line can have two directions!
That is right--vectors in one direction are counted +, in the other –
The questions below are just quickies: the teacher can add more serious ones.
-- Your ship can make 10 miles per hour but the river flows at 5 mph. What is your speed relative to the shore going (a) upstream (b) downstream?
--You run at 5 mph on a treadmill but get nowhere. Why?
Because the tread is moving in the opposite direction at 5 mph. The total velocity is therefore zero.
-- Your airplane flies north at 120 mph, while a wind blows from the west at 50 mph. What is your "ground speed" V, relative to the land below?
V2 = 122 + 52 = 14400 + 2500 = 16900. V = 130 mph.
-- Could you find the angle your path makes relative to the north direction?
Call the angle x: tan x = 5/12 = 0.41667
using the "tan–1" button on the calculator, x = 22.62 deg.
Or if you prefer: sin x = 5/13 = 0.384615, using "sin–1" , same result.
--Suppose you are given a vector in the plane (on a sheet of paper, on the map, etc.) What does it mean to resolve it into its components"?
To represent it as the sum of two other vectors--usually, in prescribed directions.
--Why would we want to do that--say, to find the ground speed of an airplane, in an actual situation?
Because the directions of the air speed and wind speed may have odd angles.
Rather than deal with those angles, it is easier to resolve each into a north-south and an east-west component, add up the components in each direction (like numbers) and then form the sum again.
--An airplane flies at 120 mph in a direction 17.13° westward of north (towards the north-west). The wind blows at 50 mph towards the south-east, 45° off the due-east direction. In what direction does the airplane move, and how fast?
Due north at 79.32 mph. Let V be the airplane's velocity, W the wind's velocity, and let us resolve these vectors in an (x,y) system with the x-axis pointing due east and the y axis due north. The components are:
Vx=-120 sin 17.13°= -35.36 Vy=120 cos 17.32° = 114.68
Wx= 50 cos 45° = 35.36 Wy= -50 sin 45°= - 35.36
The x-components cancel, the total y-component is
114.68-35.36 = 79.32
When a ball is thrown, or a shell is fired, its motion is also the superposition of two motions, as was discussed in "How Things Fall".
--Let us turn the customary (x,y) axes clockwise by 90°, so that down is the x direction, and perpendicular to it, to the right, is the y direction.
(Draw on the board). That means, downward x velocities are positive and an initial x-velocity u is negative if directed upwards.
Also the initial horizontal velocity w is positive when directed to the right
We can calculate the velocity of each motion:
Vx = u + gt Vy = w
Together they give the velocity vector V. The displacement vector S similarly has components:
Sx = ut + (1/2)gt2 Sy = wt
--We fire a gun at 1000 m/sec upward at 45° to the ground. How far will the shell travel before hitting the ground (neglecting air resistance--actual values will be smaller). Take g = 10 m/s2.
We note (u,v) are the (vertical, horizontal) components of the initial velocity, which we can call V0. So (firing in the y direction, say)
u = -1000*sin 45° = - 707 m/s
w = 1000*cos 45° = 707 m/s
At impact, Sx = 0, so
ut + (1/2)gt2 = 0
One solution is t=0--it holds no interest, just tells us we started from ground level. Divide by t (it is not zero, so we may divide by it)
–u = (1/2)gt t = –2u/g = 141.4 sec
Sy = 99.97 km, approximately 100 km.
Air resistance may cut it down to less than half.
--Pyramid builders drag a block of stone weighing 1 ton (1000 kg) up a ramp with a 5° slope. Neglecting friction (the block moves on smooth rollers), what is the force they pull against?
They have to overcome the component of the weight parallel to the surface of the ramp, which is 1000*sin 5° = 87.15 kg.
An optional excursion into language--what is the meaning of the word "telltale," as a noun?
Dictionaries will list several meanings. One of them, used aboard sailing-boats, refers to a light ribbon or yarn, tied to a shroud (a wire holding the mast) or to a sail, to indicate the direction of the wind.
A telltale next to a sail indicates mainly if the sail is filling properly. But on a shroud it serves to point out the direction of the wind relative to the moving boat. It is that direction, not the wind's velocity relative to the water, which determines how the wind is sensed by the sail! That is the direction of the wind's velocity minus the boat's velocity, subtracted like vectors (i.e. adding a velocity vector in the opposite direction to the boat's motion).
The shift due to the boat's own velocity helped a famous British astronomer understand why positions of stars seemed to shift slightly over the year; see Saberr.htm. Wind tunnels testing designs of airplane wing models also sometimes attach telltales to them.