Motion is Relative

• We always judge motion by comparing a moving object to something else.

• The “something else” is called a frame of reference.

Motion is Relative

• During the last slide:– You didn't move at all relative to your

neighbor

– You moved about 20 kilometers due to the earth's rotation

– You moved about 1000 kilometers due to the earth's motion through space

Frame of Reference for 1-D Motion

• It’s like a number line

• It has an origin

• There is a positive direction that’s defined

• And a negative direction on the other side

0 10 20 +-10-

Distance and Displacement

• Distance: How far you travel (in some time interval)– We’ll use the symbol “d”

• Displacement: How far away you are from where you started (in some time interval)

Distance and Displacement

• Example of the difference:• I run around a 400-meter track in 60 seconds.• Distance traveled during those 60 seconds:

400 m• Displacement: 0 m (I ended up back where I

started)

Displacement

• Displacement is the change in position

• It is not the same as distance traveled

• It has a direction; in one dimension, we can tell the direction by the sign (+/-)

Rate

• Rate: how a quantity changes over time.

• Mathematically: rate = quantity/time

• Ex: hot dogs/minute (“hot dogs per minute”

Speed

• Speed is the rate of changing distance

• Speed is distance per unit time

How much time ?

How far?

Velocity

• Velocity is slightly different from speed: we use displacement instead of distance, and direction matters (more about that later)

• We’ll use “v” for either speed or velocity– pay attention to the context

• Unit: m/s (meters per second)

t

ntdisplacemevelocity

time

distspeed

.

t

dv

Average vs. Instantaneous

• Average speed is the velocity over an extended period of time (like the previous example)

• Ave. speed = total distancetotal time

• Instantaneous velocity is the velocity at an instant: same equation, but time interval would be a tiny, tiny number

Average vs. Instantaneous

• I’m driving to work 4 miles away (about 8400 m)• I stop for a doughnut• I get to work in 30 minutes (1800 s)• Average speed = d = 4.67 m/s

t• When I was getting a doughnut, my

instantaneous speed was 0 m/s• When I was driving on George Mason, my

instantaneous speed was 30 mph (13.4 m/s)

Graphing Motion

• Position vs. Time

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

• So velocity = 0P

osi

tion

t

Stationary objects

Graphing Motion

• Position changes same amount every interval

• If it moves 2m in 1st second, it will move 2m every second

Posi

tion

t

Objects with constant velocity

Graphing Motion

• The slope is the change in position/ change in time

• That’s the velocity!• KEY FINDING: Slope

of position/time graph is the velocity

• Negative slope: object is moving in the negative direction

Posi

tion

t

Objects with constant velocity

Change in position

Change in time

Average vs. Instantaneous Velocity

• Slope at any point is the instantaneous velocity

• Average velocity would be the total displacement divided by total time

Posi

tion

Here slope is 10, so v = 10m/s

Here slope is 0, so v = 0 m/s

0 2 4

20

Ave. velocity would be 20/4 = 5 m/s

Interpreting Graphs• What’s going on

here?

Posi

tion

t

• Starts in a positive position

• Moves forward with constant speed

• Stops for a while

• Goes backward with constant speed (constant negative velocity)

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

Graphing Velocity vs. Time

• For constant velocity (could be sitting still, could be moving), velocity doesn’t change

• Graph is just a flat line

Posi

tio

n

t

Velo

cit

y

t

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 Velocity

Area under the curve

• Question: What does the area under the Velocity vs. Time graph tell you?

Velocity (m/s)

Time (s) 1 2 3

4 5

4

3

2

1

0

• Answer: velocity x time = distance• (By “Area under the curve”, we mean area

between the curve and the horizontal axis)

Area under the curve

• It works for changing velocity, too!

Velocity (m/s)

Time (s) 1 2 3

4 5

4

3

2

1

0

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

meters

Careful!• Velocity has a direction (in this case,

plus or minus)

Velocity (m/s)

Time (s) 1 2 3

4 5

4

3

2

1

0

• If the curve is below the axis, count the area as negative

• Here, d = -2m + 2m = 0

This triangle: -2m

This triangle: 2m

What’s happening here?

• Getting faster and faster

• Slope increases, therefore…

• Velocity increases

Posi

tio

n

t

Velo

cit

y

t

We call it Acceleration

Acceleration Notes

• Acceleration is any change in speed or direction.

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

Acceleration Notes

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

• Acceleration = change in velocity/time interval

Velo

cit

y

t

Constant Acceleration

• Note: In this class, every motion can be broken down to a constant acceleration

Velo

cit

y

t

Constant accel

Constant accel Constant

accel

NOT Constant accel

Acceleration Notes

• Mathematically:

a = Δv = “change in velocity”v = final velocityvo = initial velocity

• Units: (m/s) or m s s2

ΔΔv = v -v = v -vvoo tt tt

Acceleration Notes

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

• a = v –vo = 19 m/s – 10 m/s = 3 m/s2

t 3s• The car accelerates at 3 m/s2.

Finding Acceleration on a Velocity Graph

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

Sp

eed

t

Slope = accel = 0

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

Positive slope, so positive acceleration

Average Speed

• If the speed is changing linearly (constant acceleration)

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

• vave = v + vo 2

t

V

Vo

Vave

Average Speed: Careful!

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

• Constant acceleration: ½ * (10 + 20) = 15 mph• If I drive 10 mph for 10 miles and 20 mph for 10

miles, what’s my average speed?• 13.3 mph! • Not constant acceleration, so not 15 mph!!!