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Vectors

How do we describe this motion?

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

Remove unnecessary detail and just concentrate on his path.

Let’s consider his motion from his high point, downwards.

To make it easier to see, let’s color code aspects of this motion.

Blue = actual path of an object

Green = horizontal motion

Red = vertical motion

Look carefully at how the green (horizontal) dots are spaced and how the red (vertical) dots are spaced. What does this spacing tell us?

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

As time goes by, what (if anything) happens to the green dot spacing? What does this mean?

As time goes by, what (if anything) happens to the red dot spacing? What does this mean?

How can we use math to describe such motion?

Allow me to introduce Despicable Me’s Vector

Vector Despicable Me

Vector’s let us break motion down into X (horizontal) and Y (vertical) components.

For three dimensional motion vectors can break motion into X, Y and Z components.

Here’s an object (the black dot) located someplace in space. Note how we can describe it’s position with just three numbers.

Coordinate Plane X Y Z vector components

Image from wikipedia, Coord_planes_color.svg

For most of this class, just to keep things simple, we’ll consider objects that move in a plane (along a flat surface,) like this:

2D XY Cartesian coordinates graph

from Wikipedia, Cartesian-coordinate-system.svg

Okay, so let’s get into the details of how vectors work, in two dimensions. Our lesson is here:

X and Y components of a vector

What happens when we add motion to motion?

In this case a student kicks a ball forward – but at the same time he was on the back of a truck moving forward at a high speed. What is the resulting motion of the ball that he kicks? We need to add both of these motions together!  The technique is called vector addition.

What is happening here?

The horizontal (sideways) component of motion is independent of the vertical component.

unknown source

unknown source

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

Apps and interactives

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

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

Applications!

Vector addition is used when sailboats sail against the wind.

See this lesson from PhysicsClassroom.com Lesson Resolution of Forces

Also see Points of Sail, from School of Sailing.

Vectors let us analyze all forms of parabolic motion, such as any kind of jump.

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

physics parabola vectors skateboard

Image from real-world-physics-problems .com

The motion of all particles from a fireworks explosion all trace parabolic paths! Vectors let us analyze this.

Related math topics

How to draw a parabola. For some of the problems that we see here in class you can do the same thing, but have the parabola face in the opposite direction.

How to create a parabola GIF

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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