kinematics - analyzing motion under one dimensional motion...magnitude(size) are called. scalar...

Kinematics - Analyzing

motion under

One Dimensional Motion

What is Kinematics?

• There are two aspects to any motion

– The movement itself

– What causes the motion or what changes the

motion, which requires that forces be considered

• Mechanics –study of motion and its causes

• Two parts to mechanics

– Kinematics- deals with the concepts that are

needed to describe motion, w/out references to

forces

– Dynamics- deals with the effect that forces have on

motion

YOU deserve a speeding ticket!

I am the LAW around here and

the LAW says that the speed limit

is 55 miles per hour!

Here is the scenario!

You wake up late and have 20 minutes to get to school and you

especially do not want to be late for physics! You decide to travel

at 65 mph in a 55 mph zone. Unfortunately, Chief Genova pulls

you over. You see, the LAW (that’s me!) states that you must pay

$10 for every mile you are over the speed limit. Therefore, you

MUST pay $100 to cover the fine. Being a law abiding citizen you

agree, and pay the fine. But as you begin to leave you hear Chief

Genova say……..

BUT WAIT!

Motion Speed

Over

Fine

On road 10 mph $100

Earth’s

Rotation

1000 mph

Our city council just recently passed a

law 3 months ago that requires us to fine

you for being on the planet Earth as it

rotates on it’s axis. Since Earth rotates on

its axis at 1000 mph you must pay ….

But as you

begin to

leave you

hear Chief

Genova

say……..

$10,000

BUT WAIT!

Motion Speed

Over

Fine

On road 10 mph $100

Earth’s

Rotation

1000 mph $10,000

Earth’s

Revolving

66,621

mph

Our city council just last month passed a

law that requires us to fine you for being

on the planet Earth as it revolves around

our Sun at a speed of 66,621 mph. Thus

your fine is…..

But as you

begin to

leave you

hear Chief

Genova

say……..

$666,210

BUT WAIT!

Motion Speed

Over

Fine

On road 10 mph $100

Earth’s

Rotation

1000 mph $10,000

Earth’s

Revolving

66,621

mph

$666,210

To Vega 44,041

mph

Our city council just last week passed a


on the planet Earth which moves towards

Vega in the constellation Lyra at a speed

of 44,041 mph. Thus your fine is…..

But as you

begin to

leave you

hear Chief

Genova

say……..

$440,410

BUT WAIT!

Motion Speed

Over

Fine

On road 10 mph $100

Earth’s

Rotation

1000 mph $10,000

Earth’s

Revolving

66,621

mph

$666,210

To Vega 44,041

mph

$440,410

Milky Way 558,900

mph

Our city council just yesterday passed a


on the planet Earth in the Milky Way

which rotates at a speed of 558,900 mph.

Thus your fine is…..

Let me total

your fine!

$5,589,000

You owe……(somehow)

Your total fine is:

$6,705,720

Now be a law abiding citizen and PAY UP!

The bottom line….Motion is

RELATIVE

• It depends completely on how you want to look at the moving object.

• Example: You are sitting in an airplane which is moving at a speed of 100 km/h and there is a fly sitting on your head.

(a) What is your speed relative to the ground?

(b) What is your speed relative to the seat you're sitting it?

(c) What is the speed of the fly relative to you?

• To describe motion of an objet, must specify the location of the object at all times= establish a frame of reference

100 km/hr

0 km/hr

0 km/hr

Scalars and Vectors

• Vectors are quantities that have both a

direction and a magnitude (size).

– Ex. 2 km, 30ο north of east

• Examples of Vectors used in Physics

– Displacement

– Velocity

– Acceleration

– Force

• Scalars are quantities that have only a

magnitude(size) are called.

Scalar Example Magnitude

Speed 20 m/s

Distance 10 m

Age 15 years

Heat 1000 calories

Displacement

• Displacement (x or y) = "Change in position" is a “vector”

• Not necessarily the total distance traveled. Displacement &

distance are entirely different concepts.

– Displacement is relative to an axis, Distance how far travel w/out a

direction

o "x" displacement means moving horizontally either right or left.

o "y" displacement means moving vertically either up or down.

o Word change is expressed using the Greek letter DELTA ( Δ ).

o Find the change ALWAYS subtract FINAL - INITIAL position

o It is therefore expressed as either Δx = xf - xi or Δy = yf – yi

o SI unit = meter, can be + or – in value

o + moving along line in same direction

o - moving along line in opposite direction

Example

• Suppose a person moves in a straight line from the

lockers(at a position x = 1.0 m) toward the physics

lab(at a position x = 9.0 m) , as shown below. What is

their displacement?

The answer is positive so the person must have

been traveling horizontally to the right.

Example

•Suppose the person turns around and goes back to where

they started? What is their displacement?

The answer is negative so the person must have been traveling

horizontally to the left

What is the DISPLACEMENT for the entire trip?

What is the total DISTANCE for the entire trip?

mxxx initialfinal 00.10.1

m1688

Time Interval

• The time it takes for a given motion to

occur is called the time interval.

– There is a difference between a time and a

time interval.

– Unlike displacements, time intervals are

always positive.

if ttt

Velocity

• Average Velocity is defined as: “The RATE at

which DISPLACEMENT changes”.

• Rate = ANY quantity divided by TIME.

• SI unit for velocity is m/s

• Velocity can be + or -,

depending on its direction

•Average SPEED = the “RATE at which DISTANCE changes”

•Speed – the distance traveled divided by the elapsed time

•SI unit for speed is m/s

•Speed indicate how fast object moves but does not

give a direction t

ds

Example

• A quarterback throws a pass to a defender on the

other team who intercepts the football. Assume the

defender had to run 50 m away from the quarterback

to catch the ball, then 15 m towards the quarterback

before he is tackled. The entire play took 8 seconds.

Let's look at the defender's average velocity:

smss

mm

t

xv /38.4

08

035

Let's look at the defender's speed:

sms

m

t

ds /125.8

8

65

“m/s” is the derived unit

for both speed and

velocity.

Slope – A basic graph model

A basic model for understanding graphs in physics is SLOPE.

Using the model - Look at the formula for velocity.

Who gets to play the role of the slope?

Who gets to play the role of the y-axis or the rise?

Who gets to play the role of the x-axis or the run?

What does all the mean? It means that if your are given a Displacement

vs. Time graph, to find the velocity of an object during specific time

intervals simply find the slope.

t

xv

Run

Riseslope

Velocity

Displacement

Time

Displacement vs. Time graph

What is the velocity of the object from 0 seconds to 3 seconds? The velocity is the slope!

Steeper slope means a faster moving object.


• What is the velocity of the object from 7 seconds to 8

seconds? Once again...find the slope!

A velocity of 0 m/s. What does this mean?

It is simple....the object has simply

stopped moving for 1 second.


• What is the velocity from 8-10 seconds? You must remember!

To find the change it is final - initial.

The answer is negative! It is no surprise,

because the slope is considered to be

negative.

This value could mean several things:

The object could be traveling WEST or

SOUTH. The object is going backwards - this

being the more likely choice!

You should also understand that the slope

does NOT change from 0-3s , 5 to 7s and 8-

10s.

This means that the object has a CONSTANT

VELOCITY or IT IS NOT ACCELERATING.

Example

• It is very important that you are able to look at a graph and explain it's motion

in great detail. These graphs can be very conceptual.

Look at the time interval t = 0 to t = 9

seconds. What does the slope do?

It increases, the velocity is increasing



No slope. The velocity is ZERO.



The slope is constant and positive. The

object is moving forwards at a constant

velocity.


seconds. What does the slope do? The slope is constant and negative.

The object is moving backwards at a

constant velocity.

• Average Acceleration- change in an object’s velocity

over time period in which the change occurs

• The RATE of CHANGE of VELOCITY

• (a) can be +, or – value =vector that shows direction

• When a & v opposite directions = object slows down

or is decelerating

• When a & v same directions = object speeds up or is

accelerating

2///seconds

ondmeters/secsmorssm

t

va

Acceleration

• Example: A Cessna Aircraft goes from 0

m/s to 60 m/s in 13 seconds. Calculate the

aircraft’s acceleration.

13

060

t

va 4.62 m/s2

Example

acceleration

• Example: The Cessna now decides to land and goes from 60 m/s to 0 m/s in 11 s. Calculate the Cessna’s acceleration ?

deceleration

11

600

t

va - 5.45 m/s2

Example

Slopea

Runt

Risev

t

va

Run

RiseSlope

Acceleration

What is the acceleration from t=0s to t=3s?

60/3= 20 m/s2

What is the acceleration from t=3 s to t=5s?

0/2 = 0 m/s2

What is the acceleration from t=8s to t=9s?

0-60/1 = -60 m/s2

Acceleration – Graphical Representation

Kinematic Symbols for

Constant Acceleration

x, y Displacement

t Time

vo Initial Velocity

v Final Velocity

a Acceleration

g Acceleration due to gravity

Kinematic Equations for

Constant Acceleration

Equation #1 v = vo + at

Equation #2 x = ½(vo +v)t

Equation #3 x=vot + 1/2at2

Equation #4 v2 = vo2 + 2ax

Kinematic #1

atvv

atvvt

vv

t

va

o

oo

Kinematic #1

• Example: A boat moves slowly out of a marina (so as to

not leave a wake) with a speed of 1.50 m/s. As soon as it

passes the breakwater, leaving the marina, it throttles up

and accelerates at 2.40 m/s2. How fast is the boat

moving after accelerating for 5 seconds?

What do I know? What do I want?

vo= 1.50 m/s v = ?

a = 2.40 m/s/s

t = 5 s

v

v

atvv o

)5)(40.2()50.1(

13. 5 m/s

Kinematic #3

2

21 attvx ox

x

x

attvx ox

)5)(40.2(2

1)5)(5.1(

21

2

2

37.5 m

Example: A boat moves slowly out of a marina (so as to not

leave a wake) with a speed of 1.50 m/s. As soon as it passes

the breakwater, leaving the marina, it throttles up and

accelerates at 2.40 m/s2. How far did the boat travel during

that time?


vo= 1.50 m/s x = ?

a = 2.40 m/s/s

t = 5 s

Kinematic #4

axvv o 222


vo= 12 m/s x = ?

a = -3.5 m/s2

V = 0 m/s

x

x

x

axvv o

7144

)5.3(2120

2

2

22

• Example: You are driving through town at 12 m/s when

suddenly a ball rolls out in front of your car. You apply the

brakes and begin decelerating at 3.5 m/s2. How far do you

travel before coming to a complete stop?

20.57 m

Common Problems Students

Have

I don’t know which equation to choose!!!

Equation Missing Variable

x

v

t

atvv o

2

21 attvx ox

axvv o 222

Kinematics for the VERTICAL

Direction

• All kinematics equations can be used to analyze one

dimensional motion in either the X direction OR the y

direction

tvvytvvX

gyvvaxvv

gttvyattvx

gtvvatvv

oyox

oyyox

oyox

oyyo

)(2/1)(2/1

22

21

21

2222

22

Example

• A pitcher throws a fastball with a velocity of 43.5 m/s. It is determined that during the windup and delivery the ball covers a displacement of 2.5 meters. This is from the point behind the body to the point of release. Calculate the acceleration during his throwing motion.


vo= 0 m/s a = ?

x = 2.5 m

v = 43.5 m/s

axvv o 222

Which variable is NOT given and

NOT asked for? TIME (t)

2

22

/45.378

)5.2(205.43

sma

a

Example

• How long does it take a car at rest to cross a

35.0 m intersection after the light turns green,

if the acceleration of the car is a constant 2.00

m/s2?


vo= 0 m/s t = ?

x = 35 m

a = 2.00 m/s/s st

t

92.5

)2(2

1)0(35 2


NOT asked for? Final Velocity

2

21 attvx ox

Final Velocity(v)

Example

• A car accelerates from 12.5 m/s to 25 m/s in

6.0 seconds. What was the acceleration?

What do I

know?

What do I

want?

vo= 12.5 m/s a = ?

v = 25 m/s

t = 6s

atvv o


NOT asked for? DISPLACEMENT

2/08.2

)6(5.1225

sma

a

Displacement (x)

Example

• A stone is dropped from the top of a cliff. It is observed to hit the ground 5.78 s later. How high is the cliff?

What do I

know?

What do I

want?

voy= 0 m/s y = ?

g = -9.8 m/s2

t = 5.78 s

2

21 gttvy oy


NOT asked for? Final Velocity

mh

my

y

7.163

7.163

)78.5(9.4)78.5)(0( 2

Final Velocity(v)

The Acceleration of Gravity

• On Earth, the acceleration caused by gravity is about 9.81 m/s2

• Since the acceleration is downward, we write a = -9.81 m/s2

• The quantity 9.81 m/s2 is abbreviated g. Thus, the acceleration of gravity is a = -g

• If an object is in FREE FALL in the VERTICAL

DIRECTION, the acceleration is due to GRAVITY.

• Applies to an object falling freely near the surface of

the Earth (or any other planet) due to gravity,

ignoring air resistance.

2/8.9 smgay •It NEVER ceases to exist •It ALWAYS works DOWN •It is NEVER zero •This is ONLY true in a vacuum (no air) •Acceleration does not depend on mass of object

Free-Fall Acceleration

• Example: A person throws a ball straight

upward into the air.

Q1: What is the Acceleration at the TOP of its

path? -9.8 m/s2

Q2: What is the VELOCITY at the TOP of its path?

ZERO

Acceleration due to Gravity

Q3: What is the magnitude(#value) and direction of the acceleration, HALF way up?

-9.8 m/s2 - ALWAYS DOWNWARD

Q4: What is the magnitude(#value) and direction of the acceleration, HALF way down?

-9.8 m/s2 - ALWAYS DOWNWARD

THE BOTTOM LINE: EVERYTHING will accelerate at -9.8 m/s2 in a VACUUM, that is any situation involving NO AIR.

Acceleration due to Gravity

Graphical Analysis

Let’s Review

smslopeRun

Rise

t

xv /

seconds

meters

VELOCITY is the SLOPE of a

distance, position, or displacement

vs. time graph.

Let’s Review

What is the slope doing?

INCREASING

What is the velocity doing?

INCREASING

Let’s Review

2/seconds

ondmeters/secsmslope

Run

Riseva

Let’s Review

Describe the acceleration

during interval A.

The acceleration or SLOPE

is constant and positive.


during interval B.

The acceleration or SLOPE

is ZERO.


during interval C.

The acceleration or SLOPE is constant and negative.

Let’s Review

What is the acceleration(slope) doing?

The acceleration is INCREASING!

Let’s Review

The slopes help us sketch the motion of other graphs!

NEW MODEL = AREA MODEL

seconds

meters x seconds meters

units of viewofpoint thefromit at Looking

AREAx

HEIGHTv

BASEt

HEIGHTxBASEAREAvtxt

xv

Does all this make sense?

mA

AbhA

50.7

)5.1)(5(

mA

bhA

30

)12)(5(2

1

2

1

mA

AbhA

50.7

)5.1)(5(

1.5

m/s

13.5 m/s

Total displacement = 7.50 + 30 = 37.5 m = Total AREA under the line.

The Area Model

How FAR did this object

travel during interval

A?

mxorArea

AbhA

100

20102

1

2

1

How FAR did this object

travel during interval

B?

mxorArea

AhbA

400

2020

How FAR did this object travel during interval c?

mxorArea

AbhA

400

20402

1

2

1

The Area Model – Acceleration

2seconds

metersseconds

seconds

meters

Areav

Heighta

Baset

HeightxBaseAreatavt

va

The Area Model

v

What is the change in

Velocity during the t=2 to

t=4 interval?

smvA

AhbA

/6

32

In summary

Graph Slope Area

x vs. t Velocity N/A

v vs. t Acceleration Displacement

a vs. t N/A Velocity

Slope Graph Area

x vs. t

v vs. t

a vs. t

t (s) t (s)

x (m) v (m/s)

area = x

t (s)

a (m/s2)

area = v

Comparing & Sketching graphs

One of the more difficult applications of graphs in physics is when given

a certain type of graph and asked to draw a different type of graph

t (s)

x (m)

t (s)

v (m/s)

List 2 adjectives to describe the SLOPE or VELOCITY

1.

2.

The slope is CONSTANT

The slope is POSITIVE

How could you translate what

the SLOPE is doing on the

graph ABOVE to the Y axis on

the graph to the right?

Example

t (s)

x (m)

1st line

•

•

2nd line

•

3rd line

•

•

The slope is constant

The slope is constant

The slope is “+”

The slope is “-”

The slope is “0”

Example – Graph Matching

t (s)

v (m/s)

t (s)

a (m/s/s)

t (s)

a (m/s/s)

t (s)

a (m/s/s)

What is the SLOPE(a) doing?

The slope is increasing

kinematics - analyzing motion under one dimensional motion...magnitude(size) are called. scalar...

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