Resource Lesson
Derivatives: Instantaneous vs Average Velocities
Printer Friendly Version
As we have already discussed, there are distinctions that need to be remembered when using the terms
distance traveled, net displacement, average speed, average velocity, instantaneous speed, and instantaneous velocity
. There are times when certain terms are synonymous and other times when they are absolutely and completely different.
A few examples to refresh your memories:
While traveling in one direction along a straight line path,
total distance traveled
and
net displacement
would be equivalent HOWEVER, if the path changed direction at any time, those two terms would then be completely different with distance representing the length of the path traveled and displacement representing the length of the straight-line vector that begins at the starting point and ends at the finishing point, independent of the path taken.
In both of these examples, the displacements are identical whereas the distance traveled between points C and D is much greater than the distance traveled between points A and B since the direction switches back and forth causing the path length to be much longer.
While traveling at a
constant speed
, the calculations of an object's constant speed,
average speed, and instantaneous speed
would all yield equivalent values HOWEVER, if the speed was varied throughout the internal then the average speed during that interval would only occasionally equal its instantaneous speed at any one moment in the interval. Obviously, the notion of a constant speed would no longer be discussed under those conditions.
Both of these graph represent position vs time graphs for objects traveling along straight-line paths. On the left, the first object travels at a constant speed beginning at x = 5 and ending at x = -6 taking approximately 5.5 seconds to complete the trip. On the right graph, a second object is traveling at a variable rate beginning at x = 1 traveling down to x = 0 then up to x = 6 and finally all the way back to x = -8 taking approximately 5 seconds to complete its trip.
Average speed and average velocity
during an interval also take on their own unique properties. Average speed represents the ratio of total distance traveled divided by the total time taken to travel that distance while average velocity represents the ratio of the object's net displacement divided by the total time required to achieve that displacement. Whenever total distance traveled and net displacement differ, average speed and average velocity will be different. Since average speed is a scalar it is always reported as a positive (or zero) value; while average velocity (being a vector quantity) can have a negative, zero, or positive value depending on how you assign positive/negative directions in setting up the problem.
On the left graph displayed above the first object's average speed (11 units/ 5.5 seconds) equals the magnitude of its average velocity (11 units / 5.5 seconds, down). While on the right graph the second object's average velocity (9 units / 5 seconds, down) is much smaller than its average speed (15 units / 5 seconds).
On the other hand, an object's
instantaneous speed
will always equal the
magnitude of its instantaneous velocity
.
Related Documents
Lab:
Labs -
A Photoelectric Effect Analogy
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Acceleration Down an Inclined Plane
Labs -
Ballistic Pendulum: Muzzle Velocity
Labs -
Coefficient of Friction
Labs -
Coefficient of Kinetic Friction (pulley, incline, block)
Labs -
Collision Pendulum: Muzzle Velocity
Labs -
Conservation of Momentum
Labs -
Cookie Sale Problem
Labs -
Flow Rates
Labs -
Freefall Mini-Lab: Reaction Times
Labs -
Freefall: Timing a Bouncing Ball
Labs -
Galileo Ramps
Labs -
Gravitational Field Strength
Labs -
Home to School
Labs -
InterState Map
Labs -
LAB: Ramps - Accelerated Motion
Labs -
LabPro: Newton's 2nd Law
Labs -
LabPro: Uniformly Accelerated Motion
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 -
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 -
Video LAB: A Gravitron
Labs -
Video Lab: Ball Bouncing Across a Stage
Labs -
Video LAB: Ball Re-Bounding From a Wall
Labs -
Video Lab: Cart Push #2 and #3
Labs -
Video Lab: Falling Coffee Filters
Labs -
Video Lab: Two-Dimensional Projectile Motion
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 -
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 -
Derivation of the Kinematics Equations for Uniformly Accelerated Motion
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 -
Monkey and the Hunter
RL -
Summary: Graph Shapes for Constant Velocity
RL -
Summary: Graph Shapes for Uniformly Accelerated Motion
RL -
SVA: Slopes and Area Relationships
RL -
Thin Rods: Center of Mass
RL -
Vector Resultants: Average Velocity
Review:
REV -
Test #1: APC Review Sheet
Worksheet:
APP -
Hackensack
APP -
The Baseball Game
APP -
The Big Mac
APP -
The Cemetary
APP -
The Golf Game
APP -
The Spring Phling
CP -
2D Projectiles
CP -
Dropped From Rest
CP -
Freefall
CP -
Non-Accelerated and Accelerated Motion
CP -
Tossed Ball
CP -
Up and Down
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 -
Charged Projectiles in Uniform Electric Fields
WS -
Chase Problems #1
WS -
Chase Problems #2
WS -
Chase Problems: Projectiles
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 -
Energy Methods: More Practice with Projectiles
WS -
Energy Methods: Projectiles
WS -
Force vs Displacement Graphs
WS -
Freefall #1
WS -
Freefall #2
WS -
Freefall #3
WS -
Freefall #3 (Honors)
WS -
Horizontally Released Projectiles #1
WS -
Horizontally Released Projectiles #2
WS -
Kinematics Along With Work/Energy
WS -
Kinematics Equations #1
WS -
Kinematics Equations #2
WS -
Kinematics Equations #3: A Stop Light Story
WS -
Lab Discussion: Gravitational Field Strength and the Acceleration Due to Gravity
WS -
Position-Time Graph "Story" Combinations
WS -
Projectiles Released at an Angle
WS -
Rotational Kinetic Energy
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
PhysicsLAB
Copyright © 1997-2019
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton