Physics 207: Lecture 2, Pg 1 Lecture 2 Goals Goals: (Highlights of Chaps. 1 2.1-2.4) Units and scales, order of magnitude calculations, significant

Physics 207: Lecture 2, Pg 1

Lecture 2GoalsGoals: (Highlights of Chaps. 1 & 2.1-2.4)

Units and scales, order of magnitude calculations, significant digits (on your own for the most part)

Distinguish between Position & Displacement

Define Velocity (Average and Instantaneous), Speed

Define Acceleration

Understand algebraically, through vectors, and graphically the relationships between position, velocity and acceleration

Perform Dimensional Analysis Dimensional Analysis


Lecture 2

Reading Assignment: Reading Assignment: For next class: Finish reading Ch. 2, read For next class: Finish reading Ch. 2, read

Chapter 3 (Vectors)Chapter 3 (Vectors)


LengthDistance Distance Length (m)Length (m)Radius of Visible Universe 1 x 1026

To Andromeda Galaxy 2 x 1022

To nearest star 4 x 1016

Earth to Sun 1.5 x 1011

Radius of Earth 6.4 x 106

Sears Tower 4.5 x 102

Football Field 1 x 102

Tall person 2 x 100

Thickness of paper 1 x 10-4

Wavelength of blue light 4 x 10-7

Diameter of hydrogen atom 1 x 10-10

Diameter of proton 1 x 10-15

See http://micro.magnet.fsu.edu/primer/java/scienceopticsu

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html


Time

IntervalInterval Time (s)Time (s)Age of Universe 5 x 1017

Age of Grand Canyon 3 x 1014

Avg age of college student 6.3 x 108

One year 3.2 x 107

One hour 3.6 x 103

Light travel from Earth to Moon 1.3 x 100

One cycle of guitar A string 2 x 10-3

One cycle of FM radio wave 6 x 10-8

One cycle of visible light 1 x 10-15

Time for light to cross a proton 1 x 10-24


Order of Magnitude Calculations / Estimates

Question: If you were to eat one french fry per second, estimate how many years would it take you to eat a linear chain of trans-fat free french fries, placed end to end, that reach from the Earth to the moon?

Need to know something from your experience: Average length of french fry: 3 inches or 8 cm, 0.08 m Earth to moon distance: 250,000 miles In meters: 1.6 x 2.5 X 105 km = 4 X 108 m 1 yr x 365 d/yr x 24 hr/d x 60 min/hr x 60 s/min = 3 x 107 sec

years 200s/yr103

s105 sec. 105.0

moon) (to 105.0m 108m 104

7

910

102

8

ff


This is a very important tool to check your work Provides a reality check (if dimensional analysis fails then no

sense in putting in the numbers)

ExampleWhen working a problem you get an expression for distance

d = v t 2 ( velocity · time2 )

Quantity on left side d L length(also T time and v m/s L / T)

Quantity on right side = L / T x T2 = L x T

Left units and right units don’t match, so answer is nonsense units and right units don’t match, so answer is nonsense

Dimensional Analysis (reality check)


Exercise 1Dimensional Analysis

The force (F) to keep an object moving in a circle can be described in terms of: velocity (v, dimension L / T) of the object mass (m, dimension M) radius of the circle (R, dimension L)

Which of the following formulas for F could be correct ?

Note: Force has dimensions of ML/T2 or kg-m / s2

RmvF

2

2

RvmF(a(a

))(b(b

))(c)(c)F = mvR


Moving between pictorial and graphical representations Example: Initially I walk at a constant speed along a line

from left to right, next smoothly slow down somewhat, then smoothly speed up, and, finally walk at the same constant speed.

1. Draw a pictorial representation of my motion by using a particle model showing my position at equal time increments.

2. Draw a graphical “xy” representation of my motion with time on the x-axis and position along the y-axis.

Need to develop quantitative method(s) for algebraically describing:

1. Position2. Rate of change in position (vs. time)3. Rate of change in the change of position (vs. time)


Tracking changes in position: VECTORS

Position

Displacement (change in position)

Velocity (change in position with time)

Acceleration


Motion in One-Dimension (Kinematics) Position / Displacement

Position is usually measured and referenced to an origin:

At time= 0 seconds Joe is 10 meters to the right of the lamp origin = lamp positive direction = to the right of the lamp position vector :

10 meters

JoeO

-x +x10 meters


Joexi

Position / Displacement One second later Joe is 15 meters to the right of the lamp Displacement is just change in position.

x = xf - xi

10 meters 15 meters

Oxf = xi + x x = xf - xi = 5 meters to the right !t = tf - ti = 1 second

xf

Δx


Speed & VelocityChanges in position vs Changes in time

) timetotal(

)ntdisplacemenet ( velocityaveraget

xv

Speed, Speed, ss, is usually just the magnitude of velocity., is usually just the magnitude of velocity. The “how fast” without the direction.

Average speed references the total distance travelledAverage speed references the total distance travelled

• Average velocity = net distance covered per total time,Average velocity = net distance covered per total time,

Active Figure 1 http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=282

) timetotal(

path along taken distancespeed averaget

s

http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=282


Representative examples of speed

Speed (m/s)Speed (m/s)Speed of light 3x108

Electrons in a TV tube 107

Comets 106

Planet orbital speeds 105

Satellite orbital speeds 104

Mach 3 103

Car 101

Walking 1Centipede 10-2

Motor proteins 10-6

Molecular diffusion in liquids 10-7


Instantaneous velocityChanges in position vs Changes in time

• Instantaneous velocity, velocity at a given instant

dtdx

txv

t

)time()ntdisplaceme(limvelocity

0

http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=230

• If a position vs. time curve then just the slope at a point

x

tif

if

t ttxx

v

0ousinstantane lim


Exercise 2 Average Velocity

x (meters)

t (seconds)

2

6

-2

4

What is the average velocity over the first 4 seconds ?

(A) -1 m/s (D) not enough information to decide.

(C) 1 m/s(B) 4 m/s

1 2 430


Average Velocity Exercise 3 What is the average velocity in the last second (t = 3 to 4) ?

A. 2 m/sB. 4 m/sC. 1 m/sD. 0 m/s

x (meters)

t (seconds)

2

6

-2

4

1 2 43


What is the instantaneous velocity at the fourth second?

(A) 4 m/s (D) not enough information to decide.

(C) 1 m/s(B) 0 m/s

Exercise 4Instantaneous Velocity

x (meters)

t (seconds)

2

6

-2

4

1 2 43

dtdxv velocity

• Instantaneous velocity, velocity at a given instant


Average Speed Exercise 5

A. 2 m/sB. 4 m/sC. 1 m/sD. 0 m/s

x (meters)

t (seconds)

2

6

-2

4

1 2 43

turning point

What is the average speed over the first 4 seconds ? Here we want the ratio: total distance travelled / time(Could have asked “what was the speed in the first 4 seconds?”)


Position, Displacement, Velocity, Accelerationtime (sec) 1 2 3 4 5 6

position 1 2 3 4 5 6 (meters)

originposition vectors

displacement vectors

tx

ttxx

vif

ifx

(avg)

1 2 3 4 5 6

velocity vectors

and, this case, there is no change in velocity

tvxtx

vvv

xi

xxx

initial final

)(

0

x

y


Acceleration Changes in a particle’s motion often involve acceleration

The magnitude of the velocity vector may change The direction of the velocity vector may change

(true even if the magnitude remains constant) Both may change simultaneously

Here, a “particle” with smoothly decreasing speed

v0 vv1

v1

01 v v vtva

avg

v1 v3 v5v2 v4

v v v v v

v0


Constant Acceleration Changes in a particle’s motion often involve acceleration

The magnitude of the velocity vector may change A particle with smoothly decreasing speed:

vv11vv00 vv33 vv55vv22 vv44

aa aa aa aa aa aa

v

t0

v(t)=v0 + a t

a t = area under curve = v (an integral)

att

tva

avg


Instantaneous Acceleration

The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero


Position, velocity & acceleration All are vectors! Cannot be used interchangeably (different units!)(e.g., position vectors cannot be added directly to velocity

vectors) But the directions can be determined

“Change in the position” vector vs. time gives the direction of the velocity vector

“Change in the velocity” vector vs. time gives the direction of the acceleration vector

Given x(t) vx(t) ax (t) Given ax (t) vx (t) x(t)

v

a


And given a constant acceleration we can integrate to get explicit v and a

x

ax

vxt

t

t

txxx avv0

221

0 a v0

ttxx xx

consta x

2

2vadt

xddt

d xx

dtdx

x v

] offunction a is [ )( txtxx


A “biology” experiment Hypothesis: Older people have slower reaction

times Distance accentuates the impact of time differences Equipment: Ruler and three volunteers

Older student Younger student Record keeper Multiple trials to minimize statistics

221

0 a v Exploiting0

ttxx xx


Assignment Recap

Reading for Tuesday’s class

» Finish Chapter 2 & all of 3 (vectors)

» And first assignment is due this Wednesday

Documents

Physics 207: Lecture 2, Pg 1 Lecture 2 Goals Goals: (Highlights of Chaps. 1 2.1-2.4) Units and scales, order of magnitude calculations, significant