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Addition and Scalar Multiplication
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Vector Addition
Addition of two vectors may be affected with the help of parallelogram law of addition. The resultant(sum) vector R’ of A’ and B’ is the diagonal of the parallelogram of A and B of adjacent sides. The triangular law of vector addition follows the parallelogram law which states that: if the tail-end of the vector is placed at the arrow-head of the vector, their resultant(sum) vector is drawn…
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Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-1| Problem 1
Vector addition obeys rules that are different from those for the addition of two scalar quantities. When you add two vectors, their directions, as well as their magnitudes, must be taken into account. This Tactics Box explains how to add vectors graphically.
Learning Goal: To practice Tactics Box 1.1 Vector Addition. Vector addition obeys rules that are different from those for the addition of two scalar quantities. When you add two vectors, their directions, as well as their magnitudes, must be taken into account. This Tactics Box explains how to add vectors graphically. To add B⃗ to A⃗ (Figure 1), perform these steps:
Draw A⃗ .
Place the…
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Introduction to Tip-to-Tail Vector Addition, Vectors and Scalars http://bit.ly/2SyWj4z
Chapter 1.3 - Velocity
Velocity
It’s all about Speed. How fast can you type?
Speed := the rate of movement or change of distance from a reference point. The distance something has traveled.
Speed is relative, until you hit Einstein's theory of relativity, then it doesn’t matter how fast you are going, nothing made of matter can go faster than the speed of light. So ...
In order to travel you have to move through time. This is important to note, due to how we write the equations for velocity / speed. We say that something is moving 55 miles per hour, or 55 / hr. It’s distance traveled with in an allotted time.
the boat is traveling 55 miles per hour
the boat is traveling 55 miles / hr
Let’s say you’re on a boat, that is going 55 miles / hr, and you’re running from the stern to the stem at a pace of 8 miles / hr. How fast are you running?
x = 55 miles / hr + 8 miles / hr x = 55 miles + 8 miles / hr x = ( 55 + 8 ) miles / hr x = ( 63 ) miles / hr x = 63 miles / hr
The general equation for speed is total distance traveled over the time taken.
speed = total distance / total time
v = d / t
Vector Addition
In Chapter 1.2, we learned that distance on a cartesian coordinate system requires an X and Y component. (I copied it here for review.)
Distance := the length of an object, from top to bottom, side to side or front to back. Distance is the number of “units of measure” between 2 points.
d = p2 - p1
When distance is calculated in a 2D plane, it requires knowledge of Pythagoras’s Theorem. In this case p1 and p2 actually are XY coordinates on a cartesian grid. Knowing the XY coordinates of p1 and p2 allows you to draw a triangle. Using that triangle, you can find the distance between p1 and p2.
d^2 = b^2 + c^2 d = sqr (b^2 + c^2)
b = (y2 - y1) c = (x2 - x1)
d = sqr ( (y2 - y1)^2 + (x2 - x1) ^2)
The equation for velocity then looks like:
v = ( sqr ( (y2 - y1)^2 + (x2 - x1) ^2) ) / t
This give us the path an object traveled between x1,y1 to x2,y2. What happens when we need to look at the affects of two or more objects that are acting on each other. For example, the runner on a ship. The ship is traveling in the same direction as the ship.
We have two velocity equations, one for the ship, and one for the runner.
Runner = 8 miles / hr
Ship = 20 miles / hr
When we begin adding the affects different objects have an each other's speed, we discuss these effects as vectors. These are arrows that can be drawn on paper to indicate how one force is acting on another.
====20====> ==8> .. 28
These lines are in a striaght line, since both the person and the boat are goin in the same direction. But what if the runner runs from stem to stern, front to back. The runners arrow would point against the boats arrow, and the resulting value would be 12 mile per hour.
====20====> <8== .. 12
Now take a bird flying in the air, against the wind. The bird is flying north at 8 miles per hour, while the wind is blowing east at 6 miles per hour. What is the birds velocity relative to the ground?
The birds direction of travel will be North East, based on it’s direction and wind speed. It’s velocity will be the distance traveled per unit time. In this case, miles per hour. To calculate this value, we need to identify our points, p1 and p2, so that we can get x1, y1 and x2, y2.
First we have the bird, flying north - a straight line - at 8 miles per hour.
======8N==>
Second, we have the wind moving - another straight line - at 6 miles per hour.
====6E==>
If you remember your algebra, this is the rise over the run. The Y axis will represent north, while the X axis will represent the run. Or, rising 8 units, while running 6 units.
y = ( a * x ) + b, where a = rise / run, or 8 / 6
Yes, I really need pictures here! =(
Now, based on our units of time, miles per hour, we know that in one period, the bird and wind are moving 8 / 6. So we can choose any two points on the cartesian coordinate system, as long as p1 and p2 are a rise over run of 8 / 6.
Let’s solve for point 1 .. P1
a = 8 / 6 x = 0 b = 0
y = a * x + b y = a * x + 0 y = (8 / 6) * x y = (8 / 6) * 0 y = 0
P1 = ( 0, 0 )
Let’s solve for point 2 .. P2
a = 8 / 6 x = 6 b = 0
y = a * x + b y = a * x + 0 y = (8 / 6) * x y = (8 / 6) * 6 y = 8
P2 = ( 6, 8 )
Whoops, was that too easy?
Now that we have P1 and P2, let’s find the resulting velocity
t = miles / hour P1 = ( 0, 0 ) P2 = ( 6, 8 )
v = ( sqr ( (y2 - y1)^2 + (x2 - x1) ^2) ) / t
v = ( sqr ( (8 - 0)^2 + (6 - 0) ^2 ) ) / t v = ( sqr ( (8)^2 + (6) ^2 ) ) / t v = ( sqr ( 64 + 36 ) ) / t v = ( sqr ( 100 ) ) / t v = ( 10 ) / t v = 10 miles / hour
When doing vector addition on a 100 different vectors, it is helpful to add all the vectors up that point in the same direction. This creates a smaller group of numbers to work with than say, 100.
Visualizing Vectors in 2 Dimensions
Old curriculum: Subject: Applications of Physics 12 IRP Curriculum Organizer: Momentum IRP PLO: B2: ... analyse collisions in two dimensions using the concepts of: conservation of momentum, conservation of energy Subject: Physics 12
IRP Curriculum Organizer: Vectors IRP PLO: B1: ... perform vector analysis in one or two dimensions
Channel: Khan Academy Host: Salman Khan Date: June 16, 2011 Length: 12:53 YouTube video description: Visualizing, adding and breaking down vectors in 2 dimensions
