How things Move

Ancient Greek philosopher and scientist Aristotle developed the earliest theory of how things move.

natural motion – motion that could maintain itself without theaid of an outside agent. (Pushing a rock off theledge, falls to the ground)

liquids falling or running downhill, air rising, flames leaping upwardAristotle believed everything was made of four elements

Fire

Air

Water

Earth

Aristotle'sPeriodicTable

“Natural Motion”(vertical)

“Violent Motion” (horizontal)

How things Move

Fire

Air

Water

Earth

Aristotle'sPeriodicTable

“Natural Motion”(vertical)

“Violent Motion” (horizontal)

• Each element has its own natural motion, and its own place that it strives to be

• Aristotle believed an objects natural motion was determined by how much of each element the object contained (rock sink in water because it contained mostly earth, wood floated because it contained mostly air)

earth moves downward because Earth’s centeris it’s natural resting place

water’s naturalresting placeis on top of earth

• Violent Motion – motion that forced objects to behave contrary to an objects natural motion, meaning an external push or pull was needed

How things Move

• Aristotle believed that all motion on Earth was either “natural” or “violent”

• Motion not on earth followed a different set of rules

• 5th element – ether (from the Greek word for to kindle or blaze) – had no weight and was unchangeable, and perfect in every way

• moon, sun, planets and stars were made of ether

celestialmotion

“perfect circles”

• ether’s natural place was in the “heavens” and it moved in perfect circles

• object’s on earth could not move the way the star’s did because they did not contain ether

• Aristotle's physics governed science until about the mid 16th century

• Popular because it reinforced religious beliefs

“………fuse another five elements…” – Wu-Tang Clan

In the seventeenth century Newton developed Calculus which changedthe way we think about motion.

Describing Motion DisplacementInstantaneous SpeedAverage Speed, velocityVelocityAcceleration

Quantities whichcharacterize motion

Displacement – the change from one position, x1 to another positionx2

if xxx Greek letter, “delta”,mathematically means,“the change in”.

Displacement is a vector quantity – a vector has both size (aka magnitude) and direction

If I start at a position of –2 m, and end ata position of 3 m, what is my displacement?

mmmmm

xxx if

52323

Average velocity the ratio of the displacement, x, that occurs during aparticular time interval, t.

Examples of speed:55 mi/hr, 20 m/s, 300 km/hr

speed – 1 piece ofinfo

Examples of velocity:55 mi/hr, due West20 m/s, straight up300 km/hr, 37 degrees East of North

North

South

EastWestvelocity – 2 pieces ofinfo

Speed – answer the question: “How fast?”Velocity – answers the questions: “How fast and in what direction am I traveling?”

Quantities which only have magnitude or size are known as scalars

Quantities that have both magnitude and direction are known as vectors

if

ifavg tt

xx

t

xv

NOTE: We usually callti the starting time andset it equal to 0, ti=0

Average Velocity:

This is a position versus time plot. From here what is the averagevelocity from t = 0 to t = 3s?

s

m

ss

mm

tt

xx

t

xv

if

ifavg 3

10312

The slope ofthis line is Vavg!

The slope of the line gives us information on the direction of the velocity?

+ slope = + displacement - slope = - displacement

Average speed, sAVG, is a scalar quantity

timetotal

distance totalAVGs

GO TO HITT QUESTION

Instantaneous Velocity and Speed

If we want to know the velocity of a particle at an instant wesimply obtain the average velocity by shrinking the time interval t closer and closer to 0.

dt

dx

t

xv

t

0

lim

start finishtimetotal

distance totalspeed average

d1d2

1

1

td

1 speed avg. 2

2

td

2 speed avg.

d3d4

d5 d6

3

3

td

4

4

td

5

5

td

6

6

td

ad infinitum

the trip is built out of an infinitely large number of points just like this one

instantaneous speed at this location

zoom

-in

Example: If a particle’s position is given by x = 4-12t+3t2, whatis its velocity at t = 1s?

tttdt

dv 6123124 2

Evaluate this as t = 1 s gives us:

s

mv 66121612

What does the “-”, negative sign mean?

This tells us the direction of the particle at time t = 1s.

Acceleration: When a particle experiences a change in velocityis undergoes an acceleration. The average acceleration over a time interval is defined as:

2s

munits

tt

vv

t

va

if

ifavg :

The instantaneous acceleration is the derivative of velocity withrespect to time:

dt

dva

2

2

dt

xd

dt

dx

dt

d

dt

dva

The acceleration of a particle at any time is the second derivativeof it’s position with respect to time. NOTE: acceleration is a vectorquantity

Typical accelerations:

Ultracentrifuge 3 x 106 m/s2

Batted baseball 3 x 104 m/s2

Bungee Jump 30 m/s2

Acceleration of gravity on Earth 9.81 m/s2

Emergency stop in a car 8 m/s2

Acceleration of gravity on Moon 1.62 m/s2

Note:Acceleration of gravity on MoonAcceleration of gravity on Earth

= 1.629.81

= 0.165 16.5 %

slope of v(t)

• velocity at any time can be found from the slope of the x(t) graph

• acceleration at any time can be found from the slope of the v(t) graph

in most cases the acceleration is constant:

Example: Car skidding,free falling objects, etc.

When the acceleration is constant, theaverage acceleration and instantaneousacceleration are equal so we have:

t

vv

tt

vv

t

vaa if

if

ifavg

Multiplying both sides by t:

atvv

vvat

if

if

NOTE: Check if this is correct, what is the finalvelocity equal to at t = 0. Vi !! YEAH!

Check this out:

Remember that:

if

ifavg tt

xx

t

xv

If we set ti = 0 and tf = t we can rewrite this to be:

t

xxv ifavg

multiplying each side by t and rearrange to xf:

tvxxxxvt avgififavg it turns out that for a particle experiencing constant acceleration:

2fi

avg

vvv

now plug this into this equation

tvv

xx fiif

2now from before: atvv if

plug this into here

22

2

21

21

22

22

22

attvxxattvxx

attvxt

atvxx

tatvv

xxtvv

xx

iifiif

ii

iif

iiif

fiif

we just call this the distance, d, traveled

2

21attvd i

next we can rearrange this equation:time, t:

atvv if to solve for,

a

vvt if now substitute this equation into here

advvvvad

a

v

a

vd

a

v

a

vv

a

v

a

v

a

vv

a

vvvva

a

vvv

a

vva

a

vvvattvd

ifif

fi

iiffifi

iiffifi

ififii

2222

22

2

21

21

21

2222

22

222

2

222

2

2

Equation: tvvdxx fiif )( 21

Do this for homework!Equation summary:

Equation Missing Quantity

vf

t

a

vi

atvv if dxx if ,2

21attvd i

advv if 222

tvvdxx fiif )( 21

2

21attvdxx fif

2

21attvdxx fif

Example: On a dry road, a car with good tires may be able to brakewith a constant deceleration of 4.9s m/s2.

How long does it take the car, initially traveling at 24.6 m/s, take to stop?

Given: a = -4.9 m/s2

vi = 24.6 m/svf = 0 m/s

Find: t = ? s

smsm

a

vvt

atvvatvv

if

ifif

5924

6240

2

.

.

Objects that undergo free fall are just a case of a particle underconstant acceleration.

Aristotelian physics had a short comingwhat is I had a rock with some weight.And I had a container of water, same size, shape and weight?

• In fact all falling objects fall at the same rate, called the acceleration of gravity (neglecting air resistance)

• Drop different objects their speed will increase at the same rate!

• Their speed will increase by ~ 10 m/s (32 ft/s) every second

Free - Fallin

Free Fall Measurement –important ratiosTime Distance Speed Total Distance

1 5 m 10 m/s 5 m

Let 1 unit of distance =

the distance the object falls during the first second.

This turns out to be 4.9 m ~ 5 m

The acceleration is uniform, g = 9.8 m/s/s ~ 10 m/s/s

1

Free Fall MeasurementTime Distance Speed Total Distance

1 5 m 10 m/s 5 m

2 15 m 20 m/s 20 m

1

3

Free Fall MeasurementTime Distance Speed Total Distance

1 5 m 10 m/s 5 m

2 15 m 20 m/s 20 m

3 25 m 30 m/s 45 m

1

3

5

Free Fall MeasurementTime Distance Speed Total Distance

1 5 m 10 m/s 5 m

2 15 m 20 m/s 20 m

3 25 m 30 m/s 45 m

4 35 m 40 m/s 80 m

1

3

5

7

Free Fall MeasurementTime Distance Speed Total Distance

1 5 m 10 m/s 5 m

2 15 m 20 m/s 20 m

3 25 m 30 m/s 45 m

4 35 m 40 m/s 80 m

5 45 m 50 m/s 125 m

1

3

5

7

9

Free Fall MeasurementTime Distance Speed Total Distance

1 5 m 10 m/s 5 m

2 15 m 20 m/s 20 m

3 25 m 30 m/s 45 m

4 35 m 40 m/s 80 m

5 45 m 50 m/s 125 m

…

…

t t2

1

3

5

7

9

speed of descent ~ time of falldistance of fall ~ (time of fall)2

1 2 3 4 5

10

20

30

40

50

Free Falling ObjectS

pe

ed

(m

ete

rs/s

eco

nd

)

Time (seconds)

time (s) speed (m/s)

1 10

2 20

3 30

4 40

5 50

2104

40

15

1050

sm

ssm

sssm

sm

ts

xy

runrise

slope

holysmokes!!that’s g

speed of descent ~ time of fall

1 2 3 4 5

0

20

40

60

80

100

120

140

Free Falling ObjectD

ista

nce

(m

ete

rs)

Time (sec)

time (s) distance (m)

1 5

2 20

3 45

4 80

5 125

distance of fall ~ (time of fall)2