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Lecture 2 (Walker: 2.1Lecture 2 (Walker: 2.1--2.3)2.3)Position, Displacement, Position, Displacement,

Speed, and VelocitySpeed, and VelocityAugust 31, 2009August 31, 2009

Some illustrations courtesy Prof. J.G. Cramer, U of Washington

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Physics Readiness TestPhysics Readiness Test• Results posted on course web page and on

bulletin board across from Thornton 118• “Pass” status -- Wait list students who passed will

get an add permit• “Fail” status -- You will be dropped from Phys

111/112. • “ALEKS” status -- You will be dropped, but can

get an add permit by achieving 80% or better proficiency in ALEKS (see course web page). Continue attending class and doing homework.

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Scalar and Vector QuantitiesScalar and Vector Quantities

• Scalar quantities are completely described by magnitude only (temperature, length,…)

• Vector quantities need both magnitude (size) and direction to completely describe them (force, displacement, velocity,…)– Represented by an arrow; the length of the arrow is

proportional to the magnitude of the vector– Head of the arrow shows the direction

• Write vector as (or sometimes v) vr

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Temperature: ScalarQuantity; specified by singlenumber giving its magnitude.

Wind Velocity: Vector Quantity; specified by its magnitude & direction.

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Chapter 2 Chapter 2 OneOne--Dimensional (1Dimensional (1--D) D) KinematicsKinematicsOne dimensional kinematics refers to

motion along a straight line.Terms we will use:• Position, distance, displacement• Speed, velocity (average and

instantaneous)• Acceleration (average and instantaneous)

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Coordinate SystemsCoordinate SystemsA coordinate system is used to describe

location, or position.A coordinate system consists of:

– a fixed reference point called the origin(e.g., metal disk in street, center of table)

– a set of axes and definition of “positive” directions (e.g., “x axis points East”)

– the units for the axes (e.g., meters)

The position of an object is its location in a coordinate system. Position is a vector quantity

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Cartesian coordinate systemCartesian coordinate system

• Also called rectangular coordinate system

• x- and y- axes• position points are

labeled (x,y)

The arrow on axis indicates the “positive”direction.

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Plane polar coordinate systemPlane polar coordinate system– origin and reference

line are noted– point is distance r from

the origin in the direction of angle θ, ccw from reference line

– position points are labeled (r,θ)

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SFSU: 37.72084N, -122.476619E

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PositionPosition• Position is defined in terms of a

frame of reference (coordinate system)

• Frame A: xi >0 and xf >0 Frame B: x’i<0 but x’f >0

Note that we use subscripts to indicate different positions:xi initial position (or x0)

xf final position; x2 position #2• Vector quantity; in 1-dim, usually

use + or - to specify direction and write as just x (no arrow)

• SI Unit for position amount: meter (m)

A

B y’

x’

O’

xi xf

xi’ xf’

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DisplacementDisplacement• Displacement measures

the change in position– Represented as ∆x (if

horizontal) or ∆y (if vertical)

– Vector quantity; + or -generally sufficient to indicate direction for 1-dimensional motion

– Sometimes write as

SI Units: Meters (m)

if xxx −=∆

xi xf

∆x

if xxx rrr−=∆

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DistanceDistanceDistance (scalar) is the total length of travel. SI unit: m

If you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi.

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Displacement vs. DistanceDisplacement vs. DistanceDisplacement is the net change in position, and has a direction (maybe just + or - in 1-D).You drive from your house to the grocery store and then to your friend’s house, your net displacement is -2.1 mi:

The distance you have traveled is 10.7 mi.

mimixxx if 1.21.20 −=−=−=∆

xixf

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Distance & Displacement?Distance & Displacement?• Distance may be, but is not necessarily, the

magnitude of the displacement

Distance(blue line)

Displacement(orange line)

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PositionPosition--time graphstime graphs

Note: position-time graph is not necessarily a straight line, eventhough the motion is along x-direction

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Average SpeedThe average speed (SI unit: m/s; scalar or vector?) is defined as the distance traveled divided by the time the trip took:

Average speed = distance / elapsed time

Is the average speed of the red car 40.0 mi/h, more than 40.0 mi/h, or less than 40.0 mi/h?

Could average speed ever be negative?

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Average VelocityAverage Velocity• Say takes time ∆t for an object to undergo a

displacement • The average velocity is rate at which the displacement

occurs

• SI Unit: m/s• It is a vector; direction will be the same as the direction

of the displacement (∆t is always positive)• + or - is sufficient direction description for 1-D motion; so

xr∆

txx

txv if

average ∆

−=

∆∆

=rrr

r

txx

txv if

average ∆

−=

∆∆

=

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Average Speed and VelocityAverage velocity = displacement / elapsed time

If you return to your starting point, your average velocity is zero.

0 0.0 m/s48.0 s 0av

xvt∆

= = =∆ −

50.0 m 0 6.25 m/s8.0 s 0run

xvt∆ −

= = =∆ −

0 50.0 m 1.25 m/s48.0 s 8.0 swalk

xvt∆ −

= = =−∆ −

8 s

48 s

t=8s

t=48s

2020

Average Speed and Velocity

Graphical Interpretation of Average Velocity:The same motion, plotted one-dimensionally and as a two dimensional x-t graph:

Average speed (0-4s) = (7m)/(4s) = 1.75 m/s

Average velocity (0-4 s) = ?

Avg. speed may be, but is not always, the magnitude of avg. velocity

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Average Speed and Velocity

Graphical Interpretation of Average Velocity:The same motion, plotted one-dimensionally and as a two dimensional x-t graph:

Average speed (0-4s) = (7m)/(4s) = 1.75 m/s

Average velocity (0-4 s) = (-2 m)/(4 s) = - 0.5 m/s

Average speed may be, but is not necessarily, the magnitude of avg. velocity

Slope of line = average velocity

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Instantaneous VelocityDefinition:

(2-4)

This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous velocity. The instantaneous velocity gives the speed and direction of motion at each instant.

What about instantaneous speed? Same as instantaneous velocity?

What is the “speedometer” in a car measuring?

txv

t ∆∆

=→∆

rr

0lim

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Graphical Interpretation of Average & Instantaneous Velocity

x1

x2

x3

12

12ttxxvavg −

−=

23

23ttxxvavg −

−=

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Instantaneous VelocityThis plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity at time t is the slope of the line tangent to the curve at t.

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Calculating Instantaneous VelocityCalculating Instantaneous Velocity

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Velocity & SlopeVelocity & SlopeThe position vs. time graph of a particle moving at constant velocity has a constant slope.

The position vs. time graph of a particle moving with a changing velocity has a changing slope.

3.0 s

4.5 m

slope = velocity = 4.5 m/3.0 s = 1.5 m/s

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Key Points of Lecture 2Key Points of Lecture 2

Before the next lecture, read Walker, 2.4 - 2.6.

Homework Assignment #2a is due at 11:00 PM on Wednesday, Sept. 2.

• Scalars and Vectors

•Coordinate Systems

•Position ( x; xi, xf ) & Displacement ( )

•Average Speed ( vavg ) & Velocity ( )

•Instantaneous Speed ( v ) and Velocity ( )

•Relation between velocity & slope of position-time plot

xr∆

vravgvr