The second half of this lesson requires easy trigonometry, specifically, the use of sine and cosine functions.
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: "Vectors," section #14 |
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.
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.
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.
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?
--Does it make any difference which of the two is added first and which second?
-- Why? (The teacher demonstrates on the board.
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. UL>
-- When do vectors add like numbers?
When they all are along the same line.
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?
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"?
--Why would we want to do that--say, to find the ground speed of an airplane, in an actual situation?
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?
Vx=-120 sin 17.13°= -35.36 Vy=120 cos 17.32° = 114.68
The x-components cancel, the total y-component is
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.
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.
u = -1000*sin 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
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?
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Author and Curator: Dr. David P. Stern
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Last updated: 12.17.2001