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Vectors

Check out the motion of this person. How can we describe the motion? There are two elements to the motion: vertical, and horizontal.

http://i.imgur.com/lGFpql6.jpg

Color code the motion

Let’s remove the visual details, and concentrate on the motions.

Blue is the path of the moving object.

Green shows the horizontal motion.

Red shows the vertical motion.

Here’s the take-away idea: The vertical, and horizontal motion are totally independent!

http://www.physicsclassroom.com/mmedia/vectors/bds.cfm

How can we describe such motion in physics?

Allow me to introduce Despicable Me’s Vector

Vector Despicable Me

Vocab: A component is one part of a larger whole.

Vectors are any quantity that have a direction and a magnitude (like force or momentum)

Vectors have X and Y components

Mathematically, the components act like shadows of the force vector on the coordinate axes.

Here we see a force vector on the (x, y) plane.

The force vector is white, the x-axis is red, the y-axis is green, the origin is white.

We force vectors with their tails at the origin.

The light is shining directly into the (x, y) plane.
We see no shadows from this view.

Below is the same scene from another viewpoint.
The light is now shining directly from above (shining straight down, parallel to the y-axis.)

Note the shadow of the vector on the x-axis:
it represents the x-component of the force vector.

Next we have the same situation except the direction of the light has changed.

The light now is shining from the right, parallel to the x-axis.

A shadow of the force vector can be seen on the y-axis.

This shadow, mathematically, is the y-component of the force vector.

We are back to a flat surface diagram below; it shows how these components can be drawn.

The black vector is the two dimensional force vector, labeled F.

The red vector is the x-component of the force vector, labeled Fx.

It is pronounced ‘F sub x’. Since ‘x’ is a subscript, it really looks like this:

The subscript’s position is often implied, as here:

The green vector is the y-component of the force vector, labeled Fy, pronounced ‘F sub y’.

The components of the force vector can also be arranged this way, forming a right triangle:

forceComponents3

The sign of the components

The x-component of the force vector can be positive or negative.

  • If it points to the right, it is positive.

  • If it points to the left, it is negative.

The y-component of the force vector can be positive or negative.

  • If it points up, it is positive.

  • If it points down, it is negative.

When right triangle trigonometry is used, use your vector diagram to decide which way the components are pointing. Then assign the correct sign to your values, as a last step in your solution.

The right triangle trigonometry as presented here will always yield positive results. It is for finding the lengths of the legs of a right triangle, as one might do in geometry.

Using sailboats

One important application of this principle is in the recreational sport of sail boating. A full lesson is presented here

Lesson 1 Vector Components

Lesson Resolution of Forces

also see Points of Sail

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The horizontal (sideways) component of motion is independent of the vertical component of the motion. This isn’t intuitive so sometimes we’ll see a motion that doesn’t match our gut instincts. For instance, what is happening here?

 

unknown source

unknown source

How to draw a parabolaHow to create a parabola GIF

The Physics of Doing an Ollie on a Skateboard, or, the Science of Why I Can’t Skate

Firing a projectile on a flat surface

PhET Projectile Motion lab!

Projectile Motion app: Galileo and Einstein

Firing a projectile on a curved surface…like the Earth!

Newton’s Mountain Canon: Galileo and Einstein

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Table of contents

Vectors and Scalars

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Addition of Vectors-Graphical Methods

Because vectors are quantities that have direction as well as magnitude, they must be added in a special way. In this Chapter, we will deal mainly with displacement vectors, for which we now use the symbol and velocity vectors, But the results will apply for other vectors we encounter later.

-Giancoli Physics, Chap 3

 

Subtraction of Vectors, and Multiplication of a Vector by a Scalar

From Giancoli Physics

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Adding Vectors by Components

Adding vectors graphically using a ruler and protractor is often not sufficiently accurate and is not useful for vectors in three dimensions.

We discuss now a more powerful and precise method for adding vectors. But do not forget graphical methods—they are useful for visualizing, for checking your math, and thus for getting the correct result.

– Giancoli Physics, Chapter 3

Consider first a vector that lies in a particular plane. It can be expressed as the sum of two other vectors, called the components of the original vector.

The components are usually chosen to be along two perpendicular directions, such as
the x and y axes. The process of finding the components is known as resolving the vector into its components.

An example is shown in Fig. 3–10; the vector could be a displacement vector that points at an angle north of east, where we have chosen the positive x axis to be to the east and the positive y axis north.

This vector is resolved into its x and y components by drawing dashed lines (AB and AC) out from the tip (A) of the vector, making them perpendicular to the x and y axes.

Then the lines 0B and 0C represent the x and y components of respectively, as shown in Fig. 3–10b.

These vector components are written and In this book we usually show vector components as arrows, like vectors, but dashed.

The scalar components, and are the magnitudes of the vector components, with units, accompanied by a positive or negative sign depending on whether they point along the positive or negative x or y axis.

As can be seen in Fig. 3–10, by the parallelogram method of adding vectors.

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Resolving a vector into its components along a chosen set of x and y axes. The components, once found, themselves represent the vector. That is, the components contain as much information as the vector itself.

To add vectors using the method of components, we need to use the trigonometric
functions sine, cosine, and tangent, which we now review.

Given any angle as in Fig. 3–11a, a right triangle can be constructed by
drawing a line perpendicular to one of its sides, as in Fig. 3–11b. The longest
side of a right triangle, opposite the right angle, is called the hypotenuse, which
we label h. The side opposite the angle is labeled o, and the side adjacent is
labeled a. We let h, o, and a represent the lengths of these sides, respectively.

 

Unit Vectors

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Vector Kinematics

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Projectile Motion

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Relative Velocity

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Lessons from PhysicsClassroom.com

Vectors – Motion and Forces in Two Dimensions

  1. Lesson 1 – Vectors: Fundamentals and Operations

    1. Vectors and Direction

    2. Vector Addition

    3. Resultants

    4. Vector Components

    5. Vector Resolution

    6. Component Addition

    7. Relative Velocity and Riverboat Problems

    8. Independence of Perpendicular Components of Motion

  2. Lesson 2 – Projectile Motion

    1. What is a Projectile?

    2. Characteristics of a Projectile’s Trajectory

    3. Horizontal and Vertical Components of Velocity

    4. Horizontal and Vertical Displacemen

    5. Initial Velocity Components

    6. Horizontally Launched Projectiles – Problem-Solving

    7. Non-Horizontally Launched Projectiles – Problem-Solving

  3. Lesson 3 – Forces in Two Dimensions

    1. Addition of Forces

    2. Resolution of Forces

    3. Equilibrium and Statics

    4. Net Force Problems Revisited

    5. Inclined Planes

    6. Double Trouble in 2 Dimensions

Vectors and Projectiles

Vector Addition

Drag a vector onto the canvas. Drag the arrowhead to change its direction. Repeat up to two more times and guess the direction of the resultant. Improve your skill at adding vectors using the head-to-tail method.

Name that Vector

The Name That Vector Interactive is a skill building tool that presents users with 12 vector addition challenges. Twenty-five vectors are displayed on a grid; each challenge involves adding three of the vectors together to determine the resultant.

Vector Guessing Game

The Vector Guessing Game will challenge learner’s understanding of adding vectors.Two random vectors are displayed and learners must decide on the size and direction of the resultant.

Vector Addition: Does Order Matter?

The Vector Addition: Does Order Matter? Investigate with this app

Projectile Simulator

The variable-rich environment of the Projectile Simulator Interactive allows a learner to explore a variety of questions associated with the trajectory of a projectile. Learners can modify the launch height, the launch angle, and the launch speed and observe the effect upon the trajectory.

The Monkey and the Zookeeper

Learning Standards

Massachusetts Science Curriculum Framework

1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).

NGSS

Science and Engineering Practices: Using Mathematics and Computational Thinking

Mathematical and computational thinking in 9–12 builds on K–8 experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions.

  • Apply techniques of algebra and functions to represent and solve scientific and engineering problems.

Although there are differences in how mathematics and computational thinking are applied in science and in engineering, mathematics often brings these two fields together by enabling engineers to apply the mathematical form of scientific theories and by enabling scientists to use powerful information technologies designed by engineers. Both kinds of professionals can thereby accomplish investigations and analyses and build complex models, which might otherwise be out of the question. (NRC Framework, 2012, p. 65)

Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions. Other applications of mathematics in science and engineering include logic, geometry, and at the highest levels, calculus…. Mathematics is a tool that is key to understanding science. As such, classroom instruction must include critical skills of mathematics. The NGSS displays many of those skills through the performance expectations, but classroom instruction should enhance all of science through the use of quality mathematical and computational thinking.

Common Core Standards for Mathematics (CCSM)

High School: Number and Quantity » Vector & Matrix Quantities. Represent and model with vector quantities.

Represent and model with vector quantities.

CCSS.MATH.CONTENT.HSN.VM.A.1
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
CCSS.MATH.CONTENT.HSN.VM.A.2
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
CCSS.MATH.CONTENT.HSN.VM.A.3
(+) Solve problems involving velocity and other quantities that can be represented by vectors.

Perform operations on vectors.

CCSS.MATH.CONTENT.HSN.VM.B.4
(+) Add and subtract vectors.
CCSS.MATH.CONTENT.HSN.VM.B.4.A
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
CCSS.MATH.CONTENT.HSN.VM.B.4.B
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
CCSS.MATH.CONTENT.HSN.VM.B.4.C
Understand vector subtraction v – w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
CCSS.MATH.CONTENT.HSN.VM.B.5
(+) Multiply a vector by a scalar.
CCSS.MATH.CONTENT.HSN.VM.B.5.A
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vxvy) = (cvxcvy).
CCSS.MATH.CONTENT.HSN.VM.B.5.B
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
American Association for the Advancement of Science

Operating with Symbols and Equations

  • Become fluent in generating equivalent expressions for simple algebraic expressions and in solving linear equations and inequalities.
  • Develop fluency operating on polynomials, vectors, and matrices using by-hand operations for the simple cases and using technology for more complex cases.
AAAS Benchmarks (American Association for the Advancement of Science)
9B9-12#2: Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly. Symbolic statements can be combined to look for values of variables that will satisfy all of them at the same time.
9B9-12#5: When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all.
12B9-12#2: Find answers to problems by substituting numerical values in simple algebraic formulas and judge whether the answer is reasonable by reviewing the process and checking against typical values.
12B9-12#3: Make up and write out simple algorithms for solving problems that take several steps.
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