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SPH3U2.2vector_algebra.notebook
1
September 26, 2013
Feb 19:35 PM
Chapter 2 Motion in 2 DimensionsKEY CONCEPTSAfter completing this chapter you will be able to • explain the difference between vectors and scalars• solve vector problems involving distance, position, and displacement• describe how to determine total displacement in two dimensions by scale diagram and by the component method• solve problems related to the horizontal and vertical components of motion of a projectile using kinematics equations ﴾determine the range, maximum height, and time of flight for a projectile’s motion﴿• assess the social and environmental impacts of a technology that applies kinematics concepts
Learning Goal: By the end of today, I will be able to calculate the displacement of an object in two dimensions using algebraic techniques.
Apr 238:53 PM
Section 6.2 Vector Addition Algebraically
AB
A
B
CAC
AB + AC = AC
SPH3U2.2vector_algebra.notebook
2
September 26, 2013
Apr 239:20 PM
The Parallelogram Law for Adding Two Vectors
A
BC
D
The parallelogram law is based on the translation properties of vectors.
A vector can be moved around in the problem as long as the magnitude and direction is maintained.
The key is that the original vectors are placed tip to tail for addition.
Apr 239:20 PM
The Triangle Law for Adding Two Vectors
A
BCThe triangle law is based on the translation properties of vectors.
A vector can be moved around in the problem as long as the magnitude and direction is maintained.
The key is that the original vectors are placed tip to tail for addition.
SPH3U2.2vector_algebra.notebook
3
September 26, 2013
Feb 19:48 PM
Demo of Adding Two Vectors
Note: the parallelogram law and triangle law have a great deal of similarities due to the geometric properties the shapes share.
Feb 19:48 PM
Demo of Adding Three Vectors
SPH3U2.2vector_algebra.notebook
4
September 26, 2013
Sep 5 1:19 PM
When the forces or vectors are added sequentially you use the Triangle Law
When the forces are applied at the same point, then you use the Parallelogram Law
Addition Vectors Guide
Sep 5 1:25 PM
Review of Trig Laws to solve for the resultant vector:
SOHCAHTOA
Sine Law
Cosine Law
(capital letters are angles, lower case are sides)
SPH3U2.2vector_algebra.notebook
5
September 26, 2013
Sep 2510:23 AM
Find the missing sides of the following triangle when given the hypotenuse and one interior angle.
13m
22.6o
a
b
Sep 2510:23 AM
Find the missing sides of the following triangle when given the hypotenuse and one interior angle.
40m
77o
a
b
SPH3U2.2vector_algebra.notebook
6
September 26, 2013
Sep 2510:23 AM
Find the resultant of adding AB and BC together.
10.8km
21.8o
AB = 10.8 km at E21.8oN
BC = 11.4 km at 15.3o
15.3o
Method 1
1. Draw the resultant vector, AC.
2. Create a large triangle for AC.
3. Solve for the sides and angles.
Method 2
1. Find the x and y componentsof the two small triangles using AB and BC
2. Compare the sum of those dimensions to the large triangle created for AC.
A
B
C
Sep 2510:42 AM
Solving Vector Problems Using Components
All 2 dimensional vectors can be broken down into an X (horizontal) component and a Y (vertical) component using the Primary Trigonometric Ratios (Soh Cah Toa).
The sum of the X parts creates the X component of the resultant vector.
The sum of the Y parts creates the Y component of the resultant vector.
Positive and Negative Directions are VERY, VERY important when using this technique.
SPH3U2.2vector_algebra.notebook
7
September 26, 2013
Feb 19:48 PM
+ x
+ x
+ y
+ x
y
Demo of Adding Three Vectors
+ x
+ x
+ y
+ x
y
Sep 2510:48 AM
SPH3U2.2vector_algebra.notebook
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September 26, 2013
Sep 2510:52 AM
Sep 2510:53 AM
SPH3U2.2vector_algebra.notebook
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September 26, 2013
Sep 2510:55 AM
Velocity of the Current
Velocity of the Boat/Swimmer Crossing the River
River
Adding TWO Velocities What does that mean?
If you have ever tried to swim or paddle across a river you have experienced the addition of two velocities.
If you tried to swim across a backyard pool to a point exactly across from you, you probably didn't have much difficulty.
If you tried to swim across a rushing river to a point exactly across from you, you probably had a very difficult time because of the current or moving water.
River crossing problems are a type of twodimensional motion problem that involve perpendicular velocity vectors. The “river crossing problem” is often first introduced in terms of boats crossing rivers, but it may also involve aircraft flying through the air, and so on. These types of problems always involve two perpendicular motions that are independent of each other.
Sep 2511:00 AM
SPH3U2.2vector_algebra.notebook
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September 26, 2013
Sep 2511:03 AM
Sep 2511:11 AM
SPH3U2.2vector_algebra.notebook
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September 26, 2013
Sep 2511:12 AM
Sep 2511:05 AM