Motion 1D

CHAPTER 2

AP PHYSICS

Graphing Motion in One Dimension

Interpret graphs of position versus time for a moving object to determine the velocity of the object

Describe in words the information presented in graphs and draw graphs from descriptions of motion

Write equations that describe the position of an object moving at constant velocity

Parts of a

Graph

X-axis

Y-axis

All axes must be labeled with

appropriate units, and values.

Position vs. Time The x-axis is always

“time”

The y-axis is always “position”

The slope of the line indicates the velocity of the object.

Slope = (y2-y1)/(x2-x1) x-x0 / t-t0 Δx / Δt

Position vs. Time

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10

Time (s)

Po

siti

on

(m

)

Uniform Motion

Uniform motion is defined as equal

displacements occurring during

successive equal time periods

Straight lines on position-time graphs

mean uniform motion.

Given below is a diagram of a ball rolling along a table. Strobe

pictures reveal the position of the object at regular intervals of time,

in this case, once each 0.1 seconds.

Notice that the ball covers an equal distance between flashes. Let's assume this

distance equals 20 cm and display the ball's behavior on a graph plotting its x-

position versus time.

The slope of this line would equal 20 cm divided by 0.1 sec or 200 cm/sec. This

represents the ball's average velocity as it moves across the table. Since the

ball is moving in a positive direction its velocity is positive. That is, the ball's

velocity is a vector quantity possessing both magnitude (200 cm/sec) and

direction (positive).

Steepness of slope on Position-

Time graph

Slope is related to velocity

Steep slope = higher velocity

Shallow slope = less velocity

Different Position. Vs. Time graphsPosition vs. Time

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10

Time (s)

Po

siti

on

(m

)

Position vs. Time

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

Time (s)

Po

siti

on

(m

)

Constant positive velocity

(zero acceleration)

Constant negative velocity

(zero acceleration)

Increasing positive velocity

(positive acceleration)

Decreasing negative velocity

(positive acceleration)

Uniform MotionAccelerated

Motion

X

t

A

B

C

A … Starts at home (origin) and goes forward slowly

B … Not moving (position remains constant as time

progresses)

C … Turns around and goes in the other direction quickly,

passing up home

During which intervals was he traveling in a positive direction?

During which intervals was he traveling in a negative direction?

During which interval was he resting in a negative location?

During which interval was he resting in a positive location?

During which two intervals did he travel at the same speed?A) 0 to 2 sec B) 2 to 5 sec C) 5 to 6 sec D)6 to 7 sec E) 7 to 9 sec F)9 to 11 sec

Graphing w/

Acceleration

x

A … Start from rest south of home; increase speed gradually

B … Pass home; gradually slow to a stop (still moving north)

C … Turn around; gradually speed back up again heading south

D … Continue heading south; gradually slow to a stop near the

starting point

t

A

B C

D

Tangent

Linest

SLOPE VELOCITY

Positive Positive

Negative Negative

Zero Zero

SLOPE SPEED

Steep Fast

Gentle Slow

Flat Zero

x

On a position vs. time graph:

Increasing &

Decreasingt

x

Increasing

Decreasing

On a position vs. time graph:

Increasing means moving forward (positive direction).

Decreasing means moving backwards (negative direction).

Concavity

t

x

On a position vs. time graph:

Concave up means positive acceleration.

Concave down means negative acceleration.

Special

Points

t

x

P

QR

Inflection Pt. P, RChange of concavity,

change of acceleration

Peak or

ValleyQ

Turning point, change of

positive velocity to

negative

Time Axis

InterceptP, S

Times when you are at

“home”, or at origin

S

Next - Graphing Velocity in One

Dimension

Determine, from a graph of velocity versus

time, the velocity of an object at a specific

time

Interpret a v-t graph to find the time at

which an object has a specific velocity

Calculate the displacement of an object

from the area under a v-t graph

Velocity vs. Time

X-axis is the

“time”

Y-axis is the

“velocity”

Slope of the

line = the

acceleration

Velocity vs. Time

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10

Time (s)

Velc

oit

y (

m/s

)

Different Velocity-time graphs

Different Velocity-time graphsVelocity vs. Time

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10

Time (s)

Velo

cit

y (

m/s

)

Velocity vs. Time

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10

Time (s)

Velo

cit

y (

m/s

)

Velocity vs. Time Horizontal lines = constant velocity

Sloped line = changing velocity

Steeper = greater change in velocity per

second

Negative slope = deceleration

Acceleration vs. Time

Time is on the x-axis

Acceleration is on

the y-axis

Shows how

acceleration

changes over a

period of time.

Often a horizontal

line.

Acceleration vs. Time

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

Time (s)

Accele

ra

tio

n (

m/s

^2

)

All 3 Graphs

t

x

v

t

a

t

Real life

a

t

v

t

Note how the v graph is pointy and the a graph skips. In real life,

the blue points would be smooth curves and the orange segments

would be connected. In our class, however, we’ll only deal with

constant acceleration.

Constant Rightward Velocity

Constant Leftward Velocity

Constant Rightward

Acceleration

Constant Leftward Acceleration

Leftward Velocity with

Rightward Acceleration

Graph Practice

Male all three graphs for the following scenario:

1. Newberry starts out north of home. At time zero he’s

driving a cement mixer south very fast at a constant speed.

2. He accidentally runs over an innocent moose crossing

the road, so he slows to a stop to check on the poor moose.

3. He pauses for a while until he determines the moose is

squashed flat and deader than a doornail.

4. Fleeing the scene of the crime, Newberry takes off again

in the same direction, speeding up quickly.

5. When his conscience gets the better of him, he slows,

turns around, and returns to the crash site.

Area Underneath v-t Graph

If you calculate the area underneath

a v-t graph, you would multiply

height X width.

Because height is actually velocity

and width is actually time, area

underneath the graph is equal to

Velocity X time or

V·t

Remember that Velocity = Δx

Δt

Rearranging, we get Δx = velocity X Δt

So….the area underneath a velocity-time graph is equal to the displacement during that time period.

Areav

t

“positive area”

“negative area”

Note that, here, the areas are about equal, so even though a

significant distance may have been covered, the displacement is

about zero, meaning the stopping point was near the starting point.

The position graph shows this as well.

t

x

Velocity vs. Time The area under a velocity time graph indicates

the displacement during that time period.

Remember that the slope of the line indicates

the acceleration.

The smaller the time units the more

“instantaneous” is the acceleration at that

particular time.

If velocity is not uniform, or is changing, the

acceleration will be changing, and cannot be

determined “for an instant”, so you can only find

average acceleration

Acceleration

Determine from the curves on a velocity-time graph both the constant and instantaneous acceleration

Determine the sign of acceleration using a v-t graph and a motion diagram

Calculate the velocity and the displacement of an object undergoing constant acceleration

Acceleration

Like speed or velocity, acceleration is a rate of change, defined as the rate of change of velocity

Average Acceleration = change in velocity

t

VVa

0

Elapsed time Units of acceleration?

Rearrangement of the equation

t

VVa

0

vtav 0

0vvta

atvv 0

Finally…

This equation is to be used to find (final)

velocity of an accelerating object. You can

use it if there is or is not a beginning

velocity

atvv 0

Next - Displacement under

Constant Acceleration

Remember that displacement under

constant velocity was

With acceleration, there is no

One single instantaneous v to use,

but we could use an average

velocity of an accelerating object.

Δx = vt or x = x0 + vt

Average velocity of an accelerating object

V = ½ (v0 + v)

Average velocity of an accelerating

object would simply be ½ of sum of

beginning and ending velocities

So…….tvvxx

vtxx

)(2/1 00

0

tvvxx )(2/1 00 Key equation

Other useful kinematic equations

attvxx 2/100 2

This equation is to be used to find

final position when there is an

initial velocity, but velocity at time

to is not known.

If no time is known, use this to find

final position….

a

vvxx

2

00

22

v2 = vo2 + 2 a (x – xo )

aka

Finding final velocity if no time is

known…

)(2 00 xxavv 2 2

The equations of importance

t

VVa

0

atvv 0

tvvxx )(2/1 00

attvxx 2/100

a

vvxx

2

00

)(2 00 xxavv

2

22

22

Practical Application

Velocity/Position/Time equations Calculation of arrival times/schedules of aircraft/trains

(including vectors)

GPS technology (arrival time of signal/distance to satellite)

Military targeting/delivery

Calculation of Mass movement (glaciers/faults)

Ultrasound (speed of sound) (babies/concrete/metals) Sonar (Sound Navigation and Ranging)

Auto accident reconstruction

Explosives (rate of burn/expansion rates/timing with det. cord)