Modeling Uniform MotionModeling Uniform Motion

We believe that mere movement We believe that mere movement is life, and that the more velocity is life, and that the more velocity

it has, the more it expresses it has, the more it expresses vitality.vitality.

-- Rabindranath Tagore-- Rabindranath Tagore

The Tortoise and the HareThe Tortoise and the Hare

Tortoise Tortoise (0.2 (0.2 mphmph))

RabbitRabbit (15 (15 mphmph))

Race is along a straight 2 mile track

Runs at a constant rate

Runs for 5 minPlays and sleeps for 9 hr 53 minFinishes race in 3 min

How far ahead of the rabbit is the tortoise when he crosses the finish line?

1 mile .75 mile .5 mile .25 mile

Linear (Straight Line) Motion – Model Linear (Straight Line) Motion – Model #1#1

The simplest motion model is defined as the total distance traveled divided by the elapsed transit time. This quantity is called the average speed of an object.

distance

elapsed timerateof linear motion average speed

The mathematical form of the average speed model The mathematical form of the average speed model is:is:

By By choosingchoosing ttinitialinitial to be zero at the start of to be zero at the start of allall time time measurements, this expression simplifies to:measurements, this expression simplifies to:

The SI units associated with speed are:The SI units associated with speed are:

avg

dv

t

distance

elapsed time

length m

time s

final initialt t t

avg

dv

t

means “change in”

Speed is always a positive value because distance and time are always positive!

NOTE: ‘fast’ and ‘slow’ are relative terms used primarily when comparing one objects rate of motion to another.

ExEx..A bird travels A bird travels fasterfaster than a snail, but than a snail, but slowerslower than a bullet than a bullet..

ExEx..Previously, we determined that car B was traveling “faster” than car

A. Assuming that the distances shown below were both measured during 2 s time intervals, find the average speed of each car to support or discount our previous conclusion.

sm

avg s

mv 5.7

2

15

15 m

25 m

2512.5

2m

avg s

mv

s

AA BBB is traveling“faster”.

Ex.A car travels at a constant speed of 55 mph. How

far will it travel in 3 hrs?

NOTE: If the speed is constant, then vconstant = vavg

avgd v t

55 3mi

d hrhr

165d mi

Ex. If you travel from Arkadelphia to Little Rock in ¾ hr (a

distance of ~60 mi), what is your average speed in m/s?

Method IMethod I Method IIMethod II

avg

dv

t

6080

.75avg

miv mph

hr

.447

801

35.76

msmph

mph

m

s

160960 96540

1

md mi m

mi

3600.75 2700

1

st hr s

hr

96540

27035.

076avg

mv

s

m

s

Converting Units UpfrontConverting Units at the End

The Tortoise and the Hare (The Tortoise and the Hare (ContinuedContinued))

Time it took Tortoise to complete the race:

Distance the rabbit covers in 10 hrs:

2

.2

d milestv mph

10t hr

(15 )(5min) (0 )(9 53min) (15 )(2min)d mph mph hr mph

1 1(15 ) (15 )

12 30d mph hr mph hr

1.75d miles The rabbit is a quarter mile (0.25) behind!

Linear (Straight Line) Motion – Model #2Linear (Straight Line) Motion – Model #2

Average velocityAverage velocity

The mathematical form of our new velocity The mathematical form of our new velocity model is:model is:

The SI units that are associated with velocity The SI units that are associated with velocity are:are:

net displacement

elapsed timedirectional rateof linear motion average velocity

avg avg

xv or

t t

x

v

displacement

elapsed time

length

tim

m

e s

WarningWarning:: The The unitsunits for for speedspeed and and velocityvelocity are exactly are exactly the same. This results from both models the same. This results from both models measuring a measuring a change in positionchange in position (length) (length) divided by timedivided by time..

Unlike Unlike SpeedSpeed which is which is alwaysalways a positive, a positive, velocityvelocity can becan be negative negative ifif the displacement is in a the displacement is in a negative direction [negative direction [due to the choice of coordinate due to the choice of coordinate

systemsystem].].

Since velocity is a vector quantity, we can Since velocity is a vector quantity, we can represent it using vector arrows: represent it using vector arrows: direction of the motion = direction of the arrow direction of the motion = direction of the arrow magnitude of the velocity = length of the arrow. magnitude of the velocity = length of the arrow.

Warning:Warning:To specify a To specify a velocityvelocity numerically, a number numerically, a number

ANDAND a direction a direction must bemust be given. If no given. If no direction is given, the value could easily be direction is given, the value could easily be misinterpreted as misinterpreted as speedspeed..

Distinguishing Between Speed & Distinguishing Between Speed & VelocityVelocity

ExEx.. A car travels East 200 A car travels East 200 mm in 8 in 8 ss and then returns and then returns due due West 70 West 70 mm in 5 in 5 ss. What is the . What is the average speedaverage speed and and average average velocityvelocity of the car? of the car?

Speed:Speed:

200 70 270

8 5

d vt

d m m m

t s s

Velocity Velocity (in 1 D)(in 1 D)::

200 70

13 0x avg

m m

t s s

xv

2700.8

132

mv

s

m

s

130

1310x avg

m m

ss v East

WW EE200 200 mm

70 70 mm130 130 mm

(~ 46.5 mph)(~ 46.5 mph)(~ 24.4 mph)(~ 24.4 mph)

Visual Model of the Event

DEMO – DEMO – Around the RoomAround the Room

In order to meet at the same place at the In order to meet at the same place at the same time, the person with the greater same time, the person with the greater distance to cover distance to cover MUSTMUST travel at a higher travel at a higher rate of motion (speed)!rate of motion (speed)!

ExEx.. A car makes a round trip of 400 m (East 200 A car makes a round trip of 400 m (East 200 mm then West then West 200 200 mm) in 16 ) in 16 ss. What is the . What is the average average speedspeed and and average average velocityvelocity of the car? of the car?

200 200 mm

200 200 mmWW EE

Speed:Speed: Velocity:Velocity:

400

1625m

s

mv

s (~ 60 mph)(~ 60 mph) 0

016

x avg

m

s

m

t s

x

v

Visual Model

ExEx.. A car travels East 200 A car travels East 200 mm in 8 in 8 ss. What is the . What is the average average speedspeed and and average velocityaverage velocity of the car? of the car?

200 200 mmWW EE

Speed:Speed: Velocity:Velocity:

200

825s

mv

s

m

(~ 60 mph)(~ 60 mph)

200

825x avg

m

t s

m

s

x

v East

(~ 60 mph)(~ 60 mph)

SpeedSpeed and and velocityvelocity can can ONLYONLY have the same magnitude if the have the same magnitude if the distancedistanceand and displacementdisplacement have the same magnitude. have the same magnitude.

Visual Model

Real world example in the difference between speed and velocity

Physics Today

Sept. 2003

Pgs. 13-14

Head StartHead Start

Linear (Straight Line) Motion – Model #3Linear (Straight Line) Motion – Model #3

Average accelerationAverage acceleration

The mathematical form of our acceleration The mathematical form of our acceleration model is:model is:

The SI units that are associated with velocity The SI units that are associated with velocity are:are:

change in velocity

elapsed timechangein motion average acceleration

avg avg

va or

t t

v

a

2

velocity

elapsed time/ /

lengthtime m

m s s oi e sm

rt

ExEx.. A car changes its velocity (accelerates) from 0 A car changes its velocity (accelerates) from 0 mm//ss to to 60 60 mm//ss in 6 in 6 ss. What is the . What is the average average acceleration acceleration of the of the car?car?

60 0

6avg

m ms sas

210avg

ma

s

ExEx.. A car’s velocity increases 2 A car’s velocity increases 2 mm//ss every second. every second. What is What is the the average acceleration average acceleration of the of the car?car?

22avg

ma

s

ExEx.. If the velocity of the car remains constant If the velocity of the car remains constant ((doesn’t doesn’t changechange) over a given time interval, ) over a given time interval, what is the what is the average average acceleration acceleration of the car?of the car?

20avg

ma

s

CommentComment

To produce a change in speed (a scalar), you must make a change in magnitude.

To produce a change in velocity or acceleration (a vector), you must make a change in magnitude, direction or both!

SummarySummary

SpeedSpeed is a measure of the rate of motion of an object is a measure of the rate of motion of an object based on how long it takes to cover a specific based on how long it takes to cover a specific distancedistance between two points.between two points.

VelocityVelocity is a measure of the rate of motion of an object is a measure of the rate of motion of an object based on how long it takes to cover the based on how long it takes to cover the displacementdisplacement between two points.between two points.

The The unitsunits for for speedspeed and and velocityvelocity are exactly the same. are exactly the same.

AccelerationAcceleration measures how fast you are speeding up or measures how fast you are speeding up or slowing down.slowing down.

My dear brothers, take My dear brothers, take note of this: Everyone note of this: Everyone should be quick to listen, should be quick to listen, slow to speak and slow to slow to speak and slow to become angry, for man’s become angry, for man’s anger does not bring anger does not bring about the righteous life about the righteous life that God desires.that God desires.

James 1:19-20James 1:19-20