Resource Lesson
Derivation of the Kinematics Equations for Uniformly Accelerated Motion
Printer Friendly Version
This derivation is based on the properties of a velocity-time graph for uniformly accelerated motion where the
slope of the graph represents the acceleration
graph's area represents the displacement
Equation #1: slope = acceleration
Starting with the slope
where
gives us our first equation:
In this equation
a
represents the object's uniform acceleration
t
represents the interval of time (
t
_{2}
- t
_{1}
) over which the object's velocity changed
v
_{f}
represents the object's final velocity at the end of the time interval
v
_{o}
represents the object's initial velocity at the beginning of the time interval
Equation #2: rearrange equation #1 for v
_{f}
Equation #3: area = displacement
Before we use the variables from our graph, let's take a moment and remember from geometry the formula for the area of a trapezoid.
On our graph, this trapezoid is turned over on its side and looks like
Substituting in the following variables
v
_{o }
for
_{ }
b
_{1}
v
_{f}
for b
_{2}
h for t
allows us to rewrite the area of a trapezoid as kinematics equation #3
Equation #4: multiply equation #1 by equation #3
Equation #1:
Equation #3:
Equation #5: substitute equation #2 into equation #3
Equation #2:
Equation #3:
EQUATION SUMMARY (these MUST be memorized)
Equation
v
_{o}
v
_{f}
a
s
t
Related Documents
Lab:
Labs -
A Photoelectric Effect Analogy
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 -
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 -
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 © 1997-2021
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton