Physics 207: Lecture 2, Pg 1 Lecture 2 Goals Goals: (Highlights of Chaps. 1 2.1-2.4) Units and...
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Transcript of Physics 207: Lecture 2, Pg 1 Lecture 2 Goals Goals: (Highlights of Chaps. 1 2.1-2.4) Units and...
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
Physics 207: Lecture 2, Pg 2
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)
Physics 207: Lecture 2, Pg 3
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
Physics 207: Lecture 2, Pg 4
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
Physics 207: Lecture 2, Pg 9
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
Physics 207: Lecture 2, Pg 12
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)
Physics 207: Lecture 2, Pg 13
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
Physics 207: Lecture 2, Pg 16
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)
Physics 207: Lecture 2, Pg 17
Tracking changes in position: VECTORS
Position
Displacement (change in position)
Velocity (change in position with time)
Acceleration
Physics 207: Lecture 2, Pg 18
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
Physics 207: Lecture 2, Pg 19
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
Physics 207: Lecture 2, Pg 20
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
Physics 207: Lecture 2, Pg 21
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
Physics 207: Lecture 2, Pg 22
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
Physics 207: Lecture 2, Pg 23
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
Physics 207: Lecture 2, Pg 24
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
Physics 207: Lecture 2, Pg 25
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
Physics 207: Lecture 2, Pg 26
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?”)
Physics 207: Lecture 2, Pg 27
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
Physics 207: Lecture 2, Pg 28
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
Physics 207: Lecture 2, Pg 29
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
Physics 207: Lecture 2, Pg 30
Instantaneous Acceleration
The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero
Physics 207: Lecture 2, Pg 31
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
Physics 207: Lecture 2, Pg 32
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
Physics 207: Lecture 2, Pg 33
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
Physics 207: Lecture 2, Pg 34
Assignment Recap
Reading for Tuesday’s class
» Finish Chapter 2 & all of 3 (vectors)
» And first assignment is due this Wednesday