Resource Lesson
Vector Resultants: Average Velocity
Printer Friendly Version
A Graphical Approach: Head-To-Tail
We will first show you a graphical method, called "head-to-tail, to add together two concurrent vectors.
The terms
"head" and "tail"
are labeled as shown:
Concurrent
vectors share a common starting point.
As you view the animation shown below, note that the "tail" or start of the first vector is initially placed at the origin of the co-ordinate system. At its "head" you initially place a second "imaginary" co-ordinate, or reference, system, and then the "tail" of the second vector. If a third vector were to be added, another imaginary co-ordinate system would be placed at the head of the second vector and the tail of the third vector would begin there.
The green vector represents the sum of the two vectors, or the
resultant
. It begins at the origin of the original co-ordinate both ends and points towards the head of the last vector being added. The length of the resultant is called it
magnitude
, the angle that the resultant makes with the original x-axis is called its
direction
.
You would use the Pythagorean Theorem to mathematically calculate the resultant's magnitude. To determine its directional angle, θ, you would use the trig function tangent.
Let's use this method to answer some questions about a child's journey between two houses in which he ran 10 meters North and then 10 meters West.
We can readily tell that the distance he traveled was 20 meters, but what was his displacement? Was it equal to 20 meters, less than 20 meters, or greater than 20 meters?
To determine his actual displacement we need to draw a "head-to-tail" diagram of his trip and calculate his resultant.
The magnitude of his resultant vector equals the length of the hypotenuse in the diagram shown above.
As you can see, his displacement was less than the actual distance he traveled. It was 14.1 meters in a direction of 45º W of N. We know the angle must equal 45º since we are working in an isosceles right triangle.
Refer to the following information for the next six questions.
Now suppose our child took 4 seconds to get to his destination.
1. What was his average speed?
2. What was the magnitude of his average velocity?
3. Why are your answers to #1 and #2 not the same?
4. What is the direction of his average velocity?
5. If after an additional 4 seconds the child manages to retrace his steps and return to his original starting point, what total distance did he travel?
6. What was his final net displacement for the entire 8 seconds?
An Analytical Approach: x|y Charts
But what if his path had been more convoluted? How would we determine his resultant if it involved more than two "sections"?
When more than two vectors are added together, it is often more convenient to make use of an
x|y chart
as well as a graphical display.
Let's augment our original story. After the child walked 10 meters North and then 10 meters West, he decides to go 25 meters South, 8 meters further West, 7 meters back North and finally 14 meters East. Graphically his path would looks like that shown below.
When organized in an x|y chart this information would look like:
x
y
-10 m
10 m
-8 m
-25 m
14 m
7 m
- 4 meters
- 8 meters
This confirms that his final displacement has a magnitude of 8.9 meters at 243.3º.
Refer to the following information for the next four questions.
Now suppose our child took 5 minutes to cover the path outlined above and reach his final destination.
1. What was his average speed?
2. What was the magnitude of his average velocity?
3. Why are your answers to #1 and #2 not the same?
4. In what direction would he need to walk to return home?
Related Documents
Lab:
Labs -
2-Meter Stick Readings
Labs -
A Photoelectric Effect Analogy
Labs -
Acceleration Down an Inclined Plane
Labs -
Addition of Forces
Labs -
Ballistic Pendulum: Muzzle Velocity
Labs -
Circumference and Diameter
Labs -
Conservation of Momentum
Labs -
Cookie Sale Problem
Labs -
Density of a Paper Clip
Labs -
Determining the Distance to the Moon
Labs -
Determining the Distance to the Sun
Labs -
Eratosthenes' Measure of the Earth's Circumference
Labs -
Flow Rates
Labs -
Freefall Mini-Lab: Reaction Times
Labs -
Freefall: Timing a Bouncing Ball
Labs -
Galileo Ramps
Labs -
Home to School
Labs -
Indirect Measurements: Height by Measuring The Length of a Shadow
Labs -
Indirect Measures: Inscribed Circles
Labs -
Inertial Mass
Labs -
InterState Map
Labs -
Introductory Simple Pendulums
Labs -
LAB: Ramps - Accelerated Motion
Labs -
Lab: Rectangle Measurements
Labs -
LabPro: Newton's 2nd Law
Labs -
LabPro: Uniformly Accelerated Motion
Labs -
Marble Tube Launcher
Labs -
Mass of a Rolling Cart
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Monkey and the Hunter Animation
Labs -
Monkey and the Hunter Screen Captures
Labs -
Projectiles Released at an Angle
Labs -
Quantized Mass
Labs -
Ramps: Sliding vs Rolling
Labs -
Range of a Projectile
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Rube Goldberg Challenge
Labs -
Target Lab: Ball Bearing Rolling Down an Inclined Plane
Labs -
Terminal Velocity
Labs -
The Size of the Moon
Labs -
The Size of the Sun
Resource Lesson:
RL -
Accelerated Motion: A Data Analysis Approach
RL -
Accelerated Motion: Velocity-Time Graphs
RL -
Analyzing SVA Graph Combinations
RL -
Average Velocity - A Calculus Approach
RL -
Basic Trigonometry
RL -
Basic Trigonometry Table
RL -
Chase Problems
RL -
Chase Problems: Projectiles
RL -
Comparing Constant Velocity Graphs of Position-Time & Velocity-Time
RL -
Constant Velocity: Position-Time Graphs
RL -
Constant Velocity: Velocity-Time Graphs
RL -
Curve Fitting Patterns
RL -
Derivation of the Kinematics Equations for Uniformly Accelerated Motion
RL -
Derivatives: Instantaneous vs Average Velocities
RL -
Dimensional Analysis
RL -
Directions: Flash Cards
RL -
Freefall: Horizontally Released Projectiles (2D-Motion)
RL -
Freefall: Projectiles in 1-Dimension
RL -
Freefall: Projectiles Released at an Angle (2D-Motion)
RL -
Linear Regression and Data Analysis Methods
RL -
Metric Prefixes, Scientific Notation, and Conversions
RL -
Metric System Definitions
RL -
Metric Units of Measurement
RL -
Monkey and the Hunter
RL -
Potential Energy Functions
RL -
Properties of Lines
RL -
Properties of Vectors
RL -
Significant Figures and Scientific Notation
RL -
Summary: Graph Shapes for Constant Velocity
RL -
Summary: Graph Shapes for Uniformly Accelerated Motion
RL -
SVA: Slopes and Area Relationships
RL -
Vectors and Scalars
Review:
REV -
Honors Review: Waves and Introductory Skills
REV -
Physics I Review: Waves and Introductory Skills
REV -
Test #1: APC Review Sheet
Worksheet:
APP -
Hackensack
APP -
Puppy Love
APP -
The Baseball Game
APP -
The Big Mac
APP -
The Cemetary
APP -
The Dognapping
APP -
The Golf Game
APP -
The Pool Game
APP -
The Spring Phling
APP -
War Games
CP -
2D Projectiles
CP -
Dropped From Rest
CP -
Freefall
CP -
Inverse Square Relationships
CP -
Non-Accelerated and Accelerated Motion
CP -
Sailboats: A Vector Application
CP -
Satellites: Circular and Elliptical
CP -
Tensions and Equilibrium
CP -
Tossed Ball
CP -
Up and Down
CP -
Vectors and Components
CP -
Vectors and Resultants
CP -
Vectors and the Parallelogram Rule
NT -
Average Speed
NT -
Back-and-Forth
NT -
Crosswinds
NT -
Headwinds
NT -
Monkey Shooter
NT -
Pendulum
NT -
Projectile
WS -
Accelerated Motion: Analyzing Velocity-Time Graphs
WS -
Accelerated Motion: Graph Shape Patterns
WS -
Accelerated Motion: Practice with Data Analysis
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Advanced Properties of Freely Falling Bodies #3
WS -
Average Speed and Average Velocity
WS -
Average Speed Drill
WS -
Calculating Vector Resultants
WS -
Chase Problems #1
WS -
Chase Problems #2
WS -
Chase Problems: Projectiles
WS -
Circumference vs Diameter Lab Review
WS -
Combining Kinematics and Dynamics
WS -
Constant Velocity: Converting Position and Velocity Graphs
WS -
Constant Velocity: Position-Time Graphs #1
WS -
Constant Velocity: Position-Time Graphs #2
WS -
Constant Velocity: Position-Time Graphs #3
WS -
Constant Velocity: Velocity-Time Graphs #1
WS -
Constant Velocity: Velocity-Time Graphs #2
WS -
Constant Velocity: Velocity-Time Graphs #3
WS -
Converting s-t and v-t Graphs
WS -
Data Analysis #1
WS -
Data Analysis #2
WS -
Data Analysis #3
WS -
Data Analysis #4
WS -
Data Analysis #5
WS -
Data Analysis #6
WS -
Data Analysis #7
WS -
Data Analysis #8
WS -
Density of a Paper Clip Lab Review
WS -
Dimensional Analysis
WS -
Energy Methods: More Practice with Projectiles
WS -
Energy Methods: Projectiles
WS -
Force vs Displacement Graphs
WS -
Frames of Reference
WS -
Freefall #1
WS -
Freefall #2
WS -
Freefall #3
WS -
Freefall #3 (Honors)
WS -
Graphical Relationships and Curve Fitting
WS -
Horizontally Released Projectiles #1
WS -
Horizontally Released Projectiles #2
WS -
Indirect Measures
WS -
Kinematics Along With Work/Energy
WS -
Kinematics Equations #1
WS -
Kinematics Equations #2
WS -
Kinematics Equations #3: A Stop Light Story
WS -
Mastery Review: Introductory Labs
WS -
Metric Conversions #1
WS -
Metric Conversions #2
WS -
Metric Conversions #3
WS -
Metric Conversions #4
WS -
Position-Time Graph "Story" Combinations
WS -
Projectiles Released at an Angle
WS -
Properties of Lines #1
WS -
Properties of Lines #2
WS -
Rotational Kinetic Energy
WS -
Scientific Notation
WS -
Significant Figures and Scientific Notation
WS -
SVA Relationships #1
WS -
SVA Relationships #2
WS -
SVA Relationships #3
WS -
SVA Relationships #4
WS -
SVA Relationships #5
WS -
Work and Energy Practice: An Assortment of Situations
TB -
2A: Introduction to Motion
TB -
2B: Average Speed and Average Velocity
TB -
Antiderivatives and Kinematics Functions
TB -
Honors: Average Speed/Velocity
TB -
Kinematics Derivatives
TB -
Projectile Summary
TB -
Projectile Summary
TB -
Projectiles Mixed (Vertical and Horizontal Release)
TB -
Projectiles Released at an Angle
TB -
Set 3A: Projectiles
TB -
Working with Vectors
TB -
Working with Vectors
REV -
Math Pretest for Physics I
PhysicsLAB
Copyright © 1997-2014
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton