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Good morning

• You measure your new yard with a meter stick so that the dimensions are 20m±3cm by 30 meters±6cm. How do you make sure you will not be short ordering the perimeter fencing and the square meters of sod?

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Good morning part II

• You live on a square block 500m each side. You jog 3 times around the block. What distance did you run? What was your displacement?

• You travel 30m east then 40m south. What was your displacement?

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Kinematics I - Velocity

Position and Distance

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Kinematics• Is the study of motion without regard for the

forces causing the motion or …• the description of motion using:

Position (s)– The location of an object in space

– changes in position can be described by distance or displacement

Velocity (v)– how fast an object is changing its position

Acceleration (a)– how fast the velocity of an object is changing

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Position vs Distance

• Position – Must be defined relative to a reference point.

– Must be defined using both magnitude and direction.

– Position (s) is the location of an object in space

– units: m, cm, km

• Distance– Distance is the separation between any two points.

– It is a scalar quantity, defined only by magnitude.

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What is a Scalar?

• A scalar is a quantity that is described only by magnitude…such as distance. – it may have units e.g. kg (it may not)– it does not have any direction

– it may have a sign +ve or -ve

– it may be real, or integer

– The distance from Jacksonville to Baldwin is 10 kilometers.

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Displacement

• A objects displacement is defined as the separation between an object and a reference point.

• It is a vector quantity, defined by magnitude and direction.

• Displacement is the change in position of an object (d = sf - si)

si

sf

d = sf – si

displacement = dd

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What Is a Vector?

• A vector is a quantity that must be described by magnitude and direction.

• Magnitude– Size or quantity in unit of measurement– 10 meters, 5 Newtons

• Direction– Compass direction: North, Southeast– Angle: 25o from the positive x axis

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Vector– an arrow drawn to scale used to represent a

vector quantity

F–vector notation

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End of day 3

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Graphical (Tip-to-Tail ) Method

• Example 1: Add these vectors using the tip-to-tail method.

A B+

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Graphical (Tip-to-Tail) Method

C

A

B

A B C+ =

CIs the resultant

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What is a Resultant?

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How does one solve for displacement?d = sf – si

d = (x1,y1) – (x2,y2) How do you do this?

• Realize that displacement is a vector so you must decide to use either the Cartesian or polar coordinates

d

si = (xi,yi)

sf = (xf,yf)

d = sf– si

X

Y

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Describing Vectors

• Cartesian Coordinates (dx,dy)

• dx = x2 – x1 = distance in the x-direction

• dy = y2 – y1 = distance in the y-direction

• Polar Coordinates (d,)

• “How far and in which direction”2

122

12 )()( yyxxd

= measured directly from graph

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Describing Distance

leg 1 = 2 miles

leg 3 = 2 miles

leg 2 = 3 miles

N

Total DISTANCE Traveled= 2 miles + 3 miles + 2 miles

= 7 miles

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Describing Displacement

N

disp

lace

men

t vec

tor

Describing Displacement

First Method (Cartesian)3 miles East4 miles North(3, 4) miles

put ‘horizontal’ coordinate 1st

put ‘vertical’ coordinate 2nd

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Displacement Magnitude

N

disp

lace

men

t vec

tor

3 miles

4 m

iles

Second Method (Polar)1st - calculate length ofdisplacement vector

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Displacement Direction

N

disp

lace

men

t vec

tor

3 miles

4 m

iles

2nd – Calculate the angle using trigonometric relationships

hordverd1tan

1.5334tan 1milesmiles

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Displacement Vector(Polar Notation)

Describe the displacement vector by its length and direction

1.53,5milesd

N

disp

lace

men

t vec

tor

3 miles

4 m

iles

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Kinematics I - Velocity

Graphing

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Objectives for Graphing

• Measure, graph, analyze, and interpret laboratory data.

• Identify the relationships that may exist between variables (direct, inverse, and quadratic) and the graph shapes, which indicate these relationships (straight line, hyperbola, and parabola).

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GRAPHING

• Graphing is the pictorial relationship between quantities

– coordinate = any point w/in a graph.– Slope = rise / run or y / x

= means change

x = horizontal axis

y = vertical axis

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All relationships involve only two variables.All relationships involve only two variables.

• The Independent Variable – – this is the variable that you control or manipulate.

– It is almost always plotted on the X-axis.

• The Dependent Variable – – this is the variable that responds to the independent

variable.

– It is usually plotted in the Y-axis.

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Graphical Analysis

• Graphical analysis provides a tangible representation of a relationship from which you can make predictions and inferences.– Mass of a substance vs. volume– Speed of a car vs. braking distance– Amount of light vs. plant growth

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What are the rules?

• The graph has a title…• The scale of the graph is determined by the range

of the data and does not need to start at 0,0• The independent variable is plotted on the x-axis• The dependent variable is plotted on the y-axis• All variables have names and units• You construct a best fit line through the points

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Linear RelationshipsLinear Relationships

• The dependent variable varies directly with the independent variable.

• Plot is a straight line.• y x or y = kx• y = mx + b

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Parabolic (Exponential) RelationshipParabolic (Exponential) Relationship

• The dependent variable varies with the square of the independent variable.

• y x2 or y = kx2

• Plot does not appear as a straight line.

• Note that one variable changes faster than the other resulting in a curved line.

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Inverse (Hyperbolic) RelationshipsInverse (Hyperbolic) Relationships

• The dependent variable varies in the opposite direction of the independent variable.

• As x goes up, y goes down.

• y 1/x or y = k/x

• This results in a line that starts high and ends low. It can be straight or curved.

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Graphical AnalysisGraphical Analysis

S lop e A rea U n d er th e C u rve

G rap h ica l In fo rm ation

In te rp o la tion E xtrap o la tion

P red ic tion an d In fe ren ce

G rap h ica l A n a lys is

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SlopeSlope expresses the relationship between expresses the relationship between the variables.the variables.

• How y is changing as x is manipulated.

• M = Y/X

• Y = Y2-Y1

• X = X2-X1

• The units of the slope define the meaning of the slope.

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Area Under the Curve

• The shaded area is the area under the curve.

• You determine the area by adding up the areas of each triangle and square.

• Again, the units of this calculation, define what the number represents.

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Interpolation and ExtrapolationInterpolation and Extrapolation

• Interpolation is the reading or analysis (inference or prediction) of the graph from within the plotted data points.

• Extrapolation is the reading or analysis (inference or prediction) of the graph from outside of the plotted data points. This can lead to a significant source of error.

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Kinematics I - Velocity

Position-Time Graphs

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Graphing the Relationships Between s, v, & a

• s is the position

• v is the rate of change of s with time

• a is the rate of change of v with time

• consider a graph of s vs. time– s on vertical axis– time on horizontal axis– rate of change is interpreted as the slope

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Good morning III

• What is the position time characteristic for uniform motion, i.e. constant velocity

• What will it plot as?

• What is the slope???

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Position – Time for Constant Velocity

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Position – Time for Constant Velocity

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Position – Time for Acceleration

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Slope of Position-Time is Velocity

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Changes in the slope

• quick change– very steep slope

•positive slope–up and to the right

•negative slope–down and to the right

•slow change–very flat slope

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Slope Practice 1

• Calculate the velocity of the object whose graph is shown above.

• -24m/8s = -3 m/s

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Slope Practice 2

• Determine the velocity of the object whose motion is graphed at the left.

• 20m/5s= 4 m/s

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example

• Text page 52, #46,47

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Kinematics I - Velocity

Speed

Vs

Velocity

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Speed

• A scalar quantity referring to how fast an object is moving. – A fast-moving object has a high speed while a

slow-moving object has a low speed. – An object with no movement at all has a zero

speed.

• The distance traveled per unit time.

• It units are meters per second (m/s).

t

dv

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Velocity

• Velocity is a vector quantity describing displacement per unit time.

• Its units are meters per second (m/s).

• It is defined relative to a reference point.

• Velocity describes both speed and direction.• Depending on your direction, it can be

negative or positive.

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Average Speed

• Total distance divided by total time of travel; • Rate you would have to travel constantly to cover

the same distance in the same time

t

dvav

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Average Velocity• Total displacement divided by total time of travel;• Rate you would have to travel constantly to cover

the same displacement in the same time

if

if

tt

dd

t

dv

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Kinematics I - Velocity

Instantaneous Velocity

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Average and Instantaneous

• instantaneous speed – speed at any given instant in time; speedometer reading, for example

• instantaneous velocity - specifies how fast and in what direction one is moving at one particular point in time

• magnitude of instantaneous velocity is exactly the same as instantaneous speed

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From a Graph

• Instantaneous Velocity– the slope of the displacement versus time graph– Determine the average velocity– Take average between smaller and smaller intervals– Until you get to the tangent line– The slope of the tangent line is the instantaneous velocity at that point in time

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Kinematics I - Velocity

Problem Solution

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Problem Solution Guidelines• Draw a sketch

– Indicate origin and positive direction

• List the given quantities using the symbols of the equations. (di, vi, a)

• Identify the unknown• Write the general equation of kinematics you will

be using, i.e.2

21 attvdd

iiatvv if

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More Guidelines

• Isolate the unknown.• Rewrite the isolated equation using the known

quantities.• Check your units• Make sure you are answering the question asked.

– Your answer may be perfectly correct for the wrong question.

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Kinematics I - Velocity

Velocity-Time Graphs

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Velocity – Time for Constant Velocity

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Velocity – Time for Constant Acceleration

Time (s)

Velocity(m/s)

0 0

1 10

2 20

3 30

4 40

5 50

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Velocity – Time for Zero Acceleration

Time (s)

Velocity(m/s)

0 10

1 10

2 10

3 10

4 10

5 10

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Area Under Velocity-Time Graph

• The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 6 seconds.

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Area Under Velocity-Time Graph

• The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 4 seconds.

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Area Under Velocity-Time Graph

• The shaded area is representative of the distance traveled by the object during the time interval from 2 seconds to 5 seconds.

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Constant Positive Velocity

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Constant Negative Velocity

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Positive Velocity Positive Acceleration

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Positive Velocity Negative Acceleration

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Negative Velocity Negative Acceleration

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Negative Velocity Positive Acceleration

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Speeding Up & Slowing Down

Negative acceleration can mean speeding up or slowing down. The same is true with positive acceleration.

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Passing Lane – Position-Time

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Passing Lane – Velocity-Time

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Vector Addition in One Dimension

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homework

• Motion hand out

• Text page 55 do multiple choice practice test

• Visit physicspp.com