ch 2 motion in one dimension

8/13/2019 Ch 2 Motion in One Dimension

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Chapter 2

Motion in One Dimension


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Kinematics

Describes motion while ignoring the agentsthat caused the motion

For now, will consider motion in one

dimension Along a straight line

Will use the particle model

A particle is a point-like object, has mass butinfinitesimal size


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Position

The objects position isits location with respectto a chosen referencepoint Consider the point to be

the origin of a coordinatesystem

In the diagram, allow

the road sign to be thereference point


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Position-Time Graph

The position-time graph

shows the motion of the

particle (car)

The smooth curve is aguess as to what

happened between the

data points


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Motion of Car

Note the relationship

between the position of

the car and the points

on the graph Compare the different

representations of the

motion


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Data Table

The table gives the

actual data collected

during the motion of the

object (car) Positive is defined as

being to the right


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Alternative Representations

Using alternative representations is often an

excellent strategy for understanding a

problem

For example, the car problem used multiple

representations

Pictorial representation

Graphical representation Tabular representation

Goal is often a mathematical representation


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Displacement

Defined as the change in position during

some time interval

Represented as x

x xf-xi SI units are meters (m)

xcan be positive or negative

Different than distancethe length of a pathfollowed by a particle


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Distance vs. Displacement

An Example

Assume a player moves from one end of the

court to the other and back

Distance is twice the length of the court

Distance is always positive

Displacement is zero

x = xfxi= 0 since

xf= xi


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Vectors and Scalars

Vector quantities need both magnitude (size

or numerical value) and direction to

completely describe them

Will use + andsigns to indicate vector directions

Scalar quantities are completely described by

magnitude only


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Average Velocity

The average velocityis rate at which thedisplacement occurs

The x indicates motion along the x-axis

The dimensions are length / time [L/T]

The SI units are m/s Is also the slope of the line in the position

time graph

,f i

x avg

x xx

v t t


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Average Speed

Speed is a scalar quantity

same units as velocity

total distance / total time:

The speed has no direction and is always

expressed as a positive number

Neither average velocity nor average speed

gives details about the trip described

avgd

v

t


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Instantaneous Velocity

The limit of the average velocity as the time

interval becomes infinitesimally short, or as

the time interval approaches zero

The instantaneous velocity indicates what is

happening at every point of time


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Instantaneous Velocity, graph

The instantaneous

velocity is the slope of

the line tangent to the

xvs. tcurve

This would be the

green line

The light blue linesshow that as tgets

smaller, they approach

the green line


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Instantaneous Velocity,

equations

The general equation for instantaneous

velocity is

The instantaneous velocity can be positive,

negative, or zero

0limx tx dx

v t dt


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Instantaneous Speed

The instantaneous speed is the magnitude of

the instantaneous velocity

The instantaneous speed has no direction

associated with it


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Vocabulary Note

Velocity and speed will indicateinstantaneousvalues

Averagewill be used when the average

velocity or average speed is indicated


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Analysis Models

Analysis models are an important technique in the

solution to problems

An analysis model is a previously solved problem

It describes The behavior of some physical entity

The interaction between the entity and the environment

Try to identify the fundamental details of the problem and

attempt to recognize which of the types of problems youhave already solved could be used as a model for the new

problem


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Analysis Models, cont

Based on four simplification models

Particle model

System model

Rigid object

Wave


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Particle Under Constant

Velocity

Constant velocity indicates the instantaneous

velocity at any instant during a time interval is the

same as the average velocity during that time

interval vx= vx, avg

The mathematical representation of this situation is the

equation

Common practice is to let ti= 0 and the equation

becomes:xf= xi+ vxt (for constant vx)

f ix f i xx xxv or x x v t

t t


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Particle Under Constant

Velocity, Graph

The graph represents

the motion of a particle

under constant velocity

The slope of the graphis the value of the

constant velocity

The y-intercept is xi


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Average Acceleration

Acceleration is the rate of change of the

velocity

Dimensions are L/T2

SI units are m/s

In one dimension, positive and negative can

be used to indicate direction

,

x xf xix avg

f i

v v v

a t t t


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Instantaneous Acceleration

The instantaneous acceleration is the limit of

the average acceleration as tapproaches 0

The term acceleration will mean

instantaneous acceleration

If average acceleration is wanted, the word

average will be included

2

20

lim x xx t

v dv d xa t dt dt


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Instantaneous Acceleration --

graph

The slope of thevelocity-time graph isthe acceleration

The green linerepresents theinstantaneousacceleration

The blue line is the

average acceleration


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Graphical Comparison

Given the displacement-

time graph (a)

The velocity-time graph is

found by measuring the

slope of the position-timegraph at every instant

The acceleration-time graph

is found by measuring the

slope of the velocity-timegraph at every instant


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Acceleration and Velocity, 1

When an objects velocity and accelerationare in the same direction, the object is

speeding up

When an objects velocity and accelerationare in the opposite direction, the object is

slowing down


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Images are equally spaced. The car is moving withconstant positive velocity (shown by red arrows

maintaining the same size) Acceleration equals zero


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Images become closer together as time increases

Acceleration and velocity are in opposite directions

Acceleration is uniform (violet arrows maintain the samelength)

Velocity is decreasing (red arrows are getting shorter)

Positive velocity and negative acceleration


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Acceleration and Velocity, final

In all the previous cases, the acceleration

was constant

Shown by the violet arrows all maintaining the

same length

The diagrams represent motion of a particle

under constant acceleration

A particle under constant acceleration isanother useful analysis model

G f


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Graphical Representations of

Motion

Observe the graphs of the car under various

conditions

Note the relationships among the graphs

Set various initial velocities, positions andaccelerations

Ki i E i


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Kinematic Equations

summary


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Kinematic Equations

The kinematic equations can be used withany particle under uniform acceleration.

The kinematic equations may be used to

solve any problem involving one-dimensionalmotion with a constant acceleration

You may need to use two of the equations tosolve one problem

Many times there is more than one way tosolve a problem


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Kinematic Equations, specific

For constant a,

Can determine an objects velocity at any timetwhen we know its initial velocity and its

acceleration Assumes ti= 0 and tf= t

Does not give any information about

displacement

xf xi xv v a t


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For constant acceleration,

The average velocity can be expressed as

the arithmetic mean of the initial and final

velocities

,2

xi xfx avg

v vv


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This gives you the position of the particle interms of time and velocities

Doesnt give you the acceleration

,1

2f i x avg i xi fx x x v t x v v t


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Gives final position in terms of velocity and

acceleration

Doesnt tell you about final velocity

21

2f i xi x

x x v t a t


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For constant a,

Gives final velocity in terms of acceleration anddisplacement

Does not give any information about the time

2 2 2xf xi x f iv v a x x


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When a = 0

When the acceleration is zero,

vxf= vxi= vx

xf= xi+ vxt

The constant acceleration model reduces to

the constant velocity model

G hi l L k t M ti


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Graphical Look at Motion:

displacementtime curve

The slope of the curve

is the velocity

The curved line

indicates the velocity ischanging

Therefore, there is an

acceleration

G hi l L k t M ti


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velocitytime curve

The slope gives the

acceleration

The straight line

indicates a constantacceleration

G hi l L k t M ti


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The zero slope

indicates a constant

acceleration


accelerationtime curve

Graphical Motion with


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Graphical Motion with

Constant Acceleration

A change in the

acceleration affects the

velocity and position

Note especially thegraphs when a = 0


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Test Graphical Interpretations

Match a given velocity

graph with the

corresponding

acceleration graph Match a given

acceleration graph with

the corresponding

velocity graph(s)


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Galileo Galilei

15641642

Italian physicist and

astronomer

Formulated laws ofmotion for objects in

free fall

Supported heliocentric

universe


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Freely Falling Objects

A f reely fal l ing ob jectis any object movingfreely under the influence of gravity alone.

It does not depend upon the initial motion of

the object Droppedreleased from rest

Thrown downward

Thrown upward

Acceleration of Freely Falling


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Acceleration of Freely Falling

Object

The acceleration of an object in free fall is directed

downward, regardless of the initial motion

The magnitude of free fall acceleration is g= 9.80

m/s2

gdecreases with increasing altitude

gvaries with latitude

9.80 m/s2is the average at the Earths surface

The italicized gwill be used for the acceleration due togravity

Not to be confused with g for grams


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Acceleration of Free Fall, cont.

We will neglect air resistance

Free fall motion is constantly accelerated

motion in one dimension

Let upward be positive

Use the kinematic equations with ay= -g=

-9.80 m/s2


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Free Fallan object dropped

Initial velocity is zero

Let up be positive

Use the kinematic

equations Generally use y instead

of x since vertical

Acceleration is ay= -g= -9.80 m/s

2

vo= 0

a = -g

Free Fall an object thrown


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Free Fallan object thrown

downward

ay= -g= -9.80 m/s2

Initial velocity 0

With upward being

positive, initial velocitywill be negative vo0

a = -g

Free Fall object thrown


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Free Fall -- object thrown

upward

Initial velocity is upward, so

positive

The instantaneous velocity

at the maximum height is

zero ay= -g = -9.80 m/s

2

everywhere in the motion

v = 0

vo0

a = -g


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Thrown upward, cont.

The motion may be symmetrical

Then tup= tdown

Then v = -vo

The motion may not be symmetrical

Break the motion into various parts

Generally up and down


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Free Fall Example

Initial velocity at A is upward (+) and

acceleration is -g(-9.8 m/s2)

At B, the velocity is 0 and the acceleration

is -g(-9.8 m/s2)

At C, the velocity has the same magnitudeas at A, but is in the opposite direction

The displacement is50.0 m (it ends up50.0 m below its starting point)

Kinematic Equations from


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Kinematic Equations from

Calculus

Displacement equals

the area under the

velocitytime curve

The limit of the sum is a

definite integral

0lim ( )

f

in

t

xn n xtt

n

v t v t dt

Kinematic Equations General


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Kinematic EquationsGeneral

Calculus Form

0

0

xx

t

xf xi x

x

t

f i x

dva

dt

v v a dt

dxv

dtx x v dt


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Kinematic Equations

Calculus Form with ConstantAcceleration

The integration form of vfvigives

The integration form ofxfxigives

xf xi xv v a t

21

2f i xi xx x v t a t

General Problem Solving


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General Problem Solving

Strategy

Conceptualize

Categorize

Analyze

Finalize

Problem Solving


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Problem Solving

Conceptualize

Think about and understand the situation

Make a quick drawing of the situation

Gather the numerical information

Include algebraic meanings of phrases

Focus on the expected result

Think about units

Think about what a reasonable answer should be


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Problem SolvingCategorize

Simplify the problem

Can you ignore air resistance?

Model objects as particles

Classify the type of problem Substitution

Analysis

Try to identify similar problems you havealready solved What analysis model would be useful?


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Problem SolvingAnalyze

Select the relevant equation(s) to apply

Solve for the unknown variable

Substitute appropriate numbers

Calculate the results

Include units

Round the result to the appropriate number of

significant figures


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Problem SolvingFinalize

Check your result

Does it have the correct units?

Does it agree with your conceptualized ideas?

Look at limiting situations to be sure theresults are reasonable

Compare the result with those of similarproblems

Problem Solving Some Final


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Problem Solving Some Final

Ideas

When solving complex problems, you may

need to identify sub-problems and apply the

problem-solving strategy to each sub-part

These steps can be a guide for solvingproblems in this course

ch 2 motion in one dimension

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