Transcript Physics 207: Lecture 2 Notes

Physics 207, Lecture 2, Sept. 8
Goals: (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
Physics 207: Lecture 2, Pg 1
Physics 207, Lecture 2, Sept. 8
Assignments:
 For next class: Finish reading Ch. 2, read
Chapter 3 (Vectors)

Mastering Physics: HW1 Set due this
Wednesday, 9/10
Note: Slight change in late submission grading
(5%/hour drop)

Mastering Physics: HW2 available now, due
Wednesday, 9/17
Physics 207: Lecture 2, Pg 2
Length
Distance
Radius of Visible Universe
To Andromeda Galaxy
To nearest star
Earth to Sun
Radius of Earth
Sears Tower
Football Field
Tall person
Thickness of paper
Wavelength of blue light
Diameter of hydrogen atom
Diameter of proton
Length (m)
1 x 1026
2 x 1022
4 x 1016
1.5 x 1011
6.4 x 106
4.5 x 102
1 x 102
2 x 100
1 x 10-4
4 x 10-7
1 x 10-10
1 x 10-15
See http://micro.magnet.fsu.edu/primer/java/scienceopticsu
Physics 207: Lecture 2, Pg 3
Time
Interval
Age of Universe
Age of Grand Canyon
Avg age of college student
One year
One hour
Light travel from Earth to Moon
One cycle of guitar A string
One cycle of FM radio wave
One cycle of visible light
Time for light to cross a proton
Time (s)
5 x 1017
3 x 1014
6.3 x 108
3.2 x 107
3.6 x 103
1.3 x 100
2 x 10-3
6 x 10-8
1 x 10-15
1 x 10-24
Physics 207: Lecture 2, Pg 4
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
4 10 m
10
ff 
 0.5 10
2
8  10 m
8
Physics 207: Lecture 2, Pg 10
Converting between different systems of units

Useful Conversion factors:
 1 inch = 2.54 cm
1m
= 3.28 ft
 1 mile = 5280 ft
 1 mile = 1.61 km

Example: Convert miles per hour to meters per second:
mi 1 mi 5280 ft
1m
1 hr
m 1m
1 



 0.447 
hr
hr
mi
3.28 ft 3600 s
s 2 s
Physics 207: Lecture 2, Pg 11
Dimensional Analysis

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; this leads to the
GIGO paradigm)

Example
When working a problem you get the answer for distance
d = v t 2 ( velocity · time2 )
Quantity on left side = L
Quantity on right side = L / T x T2 = L x T

Left units and right units don’t match, so answer is nonsense
Physics 207: Lecture 2, Pg 13
Exercise 1
Dimensional 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
(a)
F = mvR
(b)
v
F  m 
R
2
(c)
kg-m / s2
mv2
F
R
Physics 207: Lecture 2, Pg 14
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 x-y representation of my motion
with time on the x-axis and position along the y-axis.
Can we develop a rigorous quantitative method with vectors
for algebraically describing position, rate of change in
position (vs. time), and the rate of change in the change of
position (vs. time)?

Physics 207: Lecture 2, Pg 17
Tracking changes in position: VECTORS

Position

Displacement

Velocity

Acceleration
Physics 207: Lecture 2, Pg 18
Motion in One-Dimension (Kinematics)
Position / Displacement

Position is usually measured and referenced to an origin:
10 meters
 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 :
-x
+x
10 meters
O
Joe
Physics 207: Lecture 2, Pg 19
Position / Displacement


One second later Joe is 15 meters to the right of the lamp
Displacement is just change in position.
x = xf - xi
15 meters
10 meters
xi
O
Δx
xf
Joe
xf = xi + x
x = xf - xi = 5 meters
t = tf - ti = 1 second
Physics 207: Lecture 2, Pg 20
Speed & Velocity
Changes in position vs Changes in time
•
Average velocity = net distance covered per total time,
x(net displaceme nt )
vaverage velocity  
t ( total time )


Speed, s, is usually just the magnitude of velocity.
 The “how fast” without the direction.
Average speed references the total distance travelled (scalar)
distance taken along path
saverage speed  
t ( total time )
Active Figure 1
http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=282
Physics 207: Lecture 2, Pg 21
Representative examples of speed
Speed
Speed of light
Electrons in a TV tube
Comets
Planet orbital speeds
Satellite orbital speeds
Mach 3
Car
Walking
Centipede
Motor proteins
Molecular diffusion in liquids
(m/s)
3x108
107
106
105
104
103
101
1
10-2
10-6
10-7
Physics 207: Lecture 2, Pg 22
Instantaneous velocity
Changes in position vs Changes in time
•
Instantaneous velocity, velocity at a given instant
x(displaceme nt ) dx
vvelocity  

t  0
t ( time )
dt
lim
Active Figure 2
http://www.phy.ntnu.edu.tw/ntnujava/main.php?t=230
Physics 207: Lecture 2, Pg 23
Exercise 2 Average Velocity
x (meters)
6
4
2
0
-2
1
2
3
4 t (seconds)
What is the average velocity over the first 4 seconds ?
(A) -1 m/s
(B) 4 m/s
(C) 1 m/s
(D) not enough
information to
decide.
Physics 207: Lecture 2, Pg 24
Average Velocity Exercise 3
What is the average velocity in the last second (t = 3 to 4) ?
x (meters)
6
4
2
-2
A.
B.
C.
D.
1
2
3
4 t (seconds)
2 m/s
4 m/s
1 m/s
0 m/s
Physics 207: Lecture 2, Pg 25
Exercise 4
Instantaneous Velocity
•
x (meters)
6
dx
vvelocity  
dt
4
2
-2
Instantaneous velocity,
velocity at a given instant
1
2
3
4 t (seconds)
What is the instantaneous velocity at the fourth second?
(A) 4 m/s
(B) 0 m/s
(C) 1 m/s
(D) not enough
information to
decide.
Physics 207: Lecture 2, Pg 26
Average Speed Exercise 5
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?”)
x (meters)
6
4
2
A.
B.
C.
D.
2 m/s
4 m/s
1 m/s
0 m/s
-2
1
2
3
4 t (seconds)
turning point
Physics 207: Lecture 2, Pg 27
Key point:

If the position x is known as a function of time, then
we can find both velocity v
x
x  x(t )
dx
vx 
dt
t1
t
vx
x   dt v(t )
t0
t
“Area” under the v(t) curve yields the change in position
 Algebraically, a special case, if the velocity is a constant

then
x(Δt)=v Δt + x0
Physics 207: Lecture 2, Pg 28
Motion in Two-Dimensions (Kinematics)
Position / Displacement

Amy has a different plan (top view):
N
10 meters
 At time= 0 seconds Amy is 10 meters to the right of the lamp
(East)
 origin = lamp
 positive x-direction = east of the lamp
 position y-direction = north of the lamp
-x
+x
10 meters
O
Amy
Physics 207: Lecture 2, Pg 30
Motion in Two-Dimensions (Kinematics)
Position / Displacement
+y
10 meters
-x
O
5 meters
-y

ri

rf
 At time= 1 second Amy is 10 meters
to the right of the lamp and 5 meters to
the south of the lamp



 r  Displaceme nt vector  r  r
f
i


 v
avg  r /t  Average velocity
+x

r
 v
r
avg
Physics 207: Lecture 2, Pg 31
N
Average Acceleration



The average acceleration of a particle as it moves is
defined as the change in the instantaneous velocity vector
divided by the time interval during which that change
occurs.
Note: bold fonts are vectors
The average
acceleration is a
vector quantity
directed along ∆v
a
Physics 207: Lecture 2, Pg 32
Instantaneous Acceleration

The instantaneous acceleration is the limit of the average
acceleration as ∆v/∆t approaches zero

Quick Comment: Instantaneous acceleration is a vector
with components parallel (tangential) and/or
perpendicular (radial) to the tangent of the path
(more in Chapter 6)
Physics 207: Lecture 2, Pg 33
Assignment Recap

Reading for Wednesday’s class on 9/10
» Finish Chapter 2 & all of 3 (vectors)
» And first assignment is due this Wednesday
Physics 207: Lecture 2, Pg 34