Graphing MotionGraphing Motion

Position vs. Position vs. TimeTime

Position is Position is same at every same at every time time (d = 0)(d = 0)

So speed = 0So speed = 0

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Stationary Stationary objectsobjects

Graphing MotionGraphing Motion

Position changes Position changes same amount same amount every intervalevery interval

If it moves 2m in If it moves 2m in 11stst second, it will second, it will move 2m every move 2m every secondsecond

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Objects with Objects with constant speedconstant speed

Graphing MotionGraphing Motion

The slope is the change The slope is the change in position/ change in in position/ change in timetime

That’s the speed or That’s the speed or velocity!velocity!

KEY FINDING: Slope KEY FINDING: Slope of position/time graph of position/time graph is the speed or is the speed or velocityvelocity

Negative slope: object Negative slope: object is moving in the is moving in the negative directionnegative direction

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Objects with Objects with constant constant velocityvelocity

Change in position

Change in time

Interpreting GraphsInterpreting Graphs

What’s going on here?What’s going on here?

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• Starts in a positive position

• Moves forward with constant speed

• Stops for a while

• Goes backward with constant speed

• Goes forward with constant speed to the origin (x = 0)

Graphing Speed vs. TimeGraphing Speed vs. Time For constant speed (could be sitting still, For constant speed (could be sitting still,

could be moving), speed doesn’t changecould be moving), speed doesn’t change Graph is just a flat lineGraph is just a flat line

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Case 1: No Motion

Case 2: Positive Constant Velocity

REMEMBER: This is just the slope of the position/time graph!

Case 1: No Motion

Case 2: Positive Constant Speed

Area under the curveArea under the curve Question: What does the Question: What does the areaarea

under the under the Velocity vs. Time Velocity vs. Time graph tell you?graph tell you?

Speed (m/s)

Time (s) 1 2 3

4 5

4

3

2

1

0

Answer: velocity x time = distanceAnswer: velocity x time = distance

Area under the curveArea under the curve It works for changing velocity, It works for changing velocity,

too!too!

Speed (m/s)

Time (s) 1 2 3

4 5

4

3

2

1

0

What is the total displacement?What is the total displacement? Area of the triangle: ½ * 4 * 4 = 8 Area of the triangle: ½ * 4 * 4 = 8

metersmeters

What’s happening here?What’s happening here?

Getting faster and fasterGetting faster and faster Slope increases, therefore…Slope increases, therefore… Speed increasesSpeed increases

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We call it Acceleration

Acceleration NotesAcceleration Notes

Acceleration is any Acceleration is any changechange in speed or direction.in speed or direction.

Acceleration occurs when an Acceleration occurs when an object speeds up, slows object speeds up, slows down (or changes direction– down (or changes direction– we’ll see this later)we’ll see this later)

Acceleration NotesAcceleration Notes

Uniform (or Uniform (or constant) constant) acceleration:acceleration: when when an object accelerates an object accelerates at a constant rate at a constant rate over a period of over a period of time.time.

Acceleration = change in Acceleration = change in velocity/time intervalvelocity/time interval

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Acceleration NotesAcceleration Notes

Mathematically:Mathematically:

a =a = ΔΔv = “change in v = “change in velocity”velocity”

v = final velocityv = final velocityvvoo = initial velocity = initial velocity

Units: Units: (m/s)(m/s) or or mm s s s s22

ΔΔv = v -v = v -vvoo tt tt

Acceleration NotesAcceleration Notes

Example: Example: A car starts out traveling at 10 A car starts out traveling at 10 m/s and accelerates to 19 m/s in m/s and accelerates to 19 m/s in a time of 3 seconds. What is a time of 3 seconds. What is the acceleration of the car?the acceleration of the car?

a = a = vvff –v –vii = = 19 m/s – 10 m/s19 m/s – 10 m/s = 3 = 3 m/sm/s22

tt 3s3s The car accelerates at 3 m/sThe car accelerates at 3 m/s22..

Finding Acceleration Finding Acceleration on a Velocity Graphon a Velocity Graph

For linear change in velocity, For linear change in velocity, acceleration is the slope of the acceleration is the slope of the velocity graphvelocity graph

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Slope = accel = 0

Negative slope, so neg. acceleration (sometimes called “deceleration”

Positive slope, so positive acceleration

Average SpeedAverage Speed

If the speed is changing linearly If the speed is changing linearly (constant acceleration)(constant acceleration)

Average speed is just the average of Average speed is just the average of the initial and final speedsthe initial and final speeds

vvaveave = = v + vv + voo 2 2

t

V

Vo

Vave

Average Speed: Careful!Average Speed: Careful!

If I accelerate uniformly from 10 to 20 mph If I accelerate uniformly from 10 to 20 mph (miles per hour), what’s my average speed?(miles per hour), what’s my average speed?

Constant acceleration: ½ * (10 + 20) = 15 Constant acceleration: ½ * (10 + 20) = 15 mphmph

If I drive 10 mph for 10 miles and 20 mph If I drive 10 mph for 10 miles and 20 mph for 10 miles, what’s my average speed?for 10 miles, what’s my average speed?

13.3 mph! 13.3 mph! Not constant acceleration, so not 15 mph!!!Not constant acceleration, so not 15 mph!!!