Resource Lesson
Average Velocity - A Calculus Approach
Printer Friendly Version
Motion of a particle in one dimension when given a position (displacement) function
Suppose a particle is moving back and forth along the x-axis with a position function (the coordinate giving the location of the particle on the x-axis) given by
x(t) = t
^{2}
– 2t – 3
.
Assume that
t
is in seconds, t < 0 is not possible, and the coordinates on the x-axis are in meters. We already know the following facts:
The initial position is x(0) = –3 meters which is 3 meters to the left of the origin
The coordinate on the x-axis giving location when t = 2 is x(2) = 4 – 4 – 3 = –3 meters.
The particle is at the origin when x(t) = 0 = (t – 3)(t + 1) or when t = 3 and t = –1
(we exclude the case where t = –1 since we exclude t < 0).
This position function x(t) is really a vector function called the "displacement" of the particle from the origin.
x(0) = – 3 means the particle is a distance (magnitude of the vector x) of 3 to the left (direction) of the origin.
Let's use these facts to answer the next two questions.
What is the average velocity between times t = 2 and t = 5?
What is the instantaneous velocity at t = 2?
Motion of a particle in one dimension when given a velocity or acceleration function
Suppose we know that the velocity of a particle moving back and forth on the x-axis is given by
v(t) = 2t – 2
and the initial position x(0) = –3. Notice that this is the same particle as in Part I above.
Some basic facts:
v(t) = 2t – 2
is the derivative of the position function
x(t) = t
^{2}
– 2t – 3
.
a(t) = 2
is the derivative of v(t) and is the uniform (or constant) acceleration of the particle.
v(t) = 0 when t = 1 which invites us to investigate the sign of our velocity function
v(t) < 0 when t < 1 and v(t) > 0 when t > 1.
To determine an object's position given its velocity function and an initial position we will use a basic fact from calculus:
Given the rate of change of a quantity and a time interval, the “net change” is the integral of the rate of change over the given time period.
Let's use these fact to look at three final questions.
What is the position of the particle after the first 5 seconds of motion?
What is the total distance traveled by the particle in the first 5 seconds?
Is the particle speeding up or slowing down when t = 4?
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 -
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 -
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 -
Derivatives: Instantaneous vs Average Velocities
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 © 1997-2017
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton