Download Physics 207: Lecture 2 Notes

Lecture 2
Goals: (Highlights of Chaps. 1 & 2.1-2.4)
 Conduct order of magnitude calculations,
 Determine units, scales, significant digits (in discussion or on
your own)
 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
Reading Quiz
Displacement, position, velocity & acceleration are the
main quantities that we will discuss today.
Which of these 4 quantities have the same units
A. Velocity & position
B. Velocity & acceleration
C. Acceleration & displacement
D. Position & displacement
E. Position & acceleration
Physics 207: Lecture 2, Pg 2
Perspective Length/Time/Mass
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
Universal standard: The speed of light is defined to be exactly
299 792 458 m/s and so one measures how far light travels in
1/299 792 458 of a second
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
World’s most accurate timepiece: Cesium fountain Atomic Clock
Lose or gain one second in some 138 million years
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
 1 yr x 365 d/yr x 24 hr/d x 60 min/hr x 60 s/min = 3 x 107 sec
4  108 m
10
ff 

0
.
5

10
(to moon)
2
8  10 m
9
5

10
s
10
0.5  10 sec. 
 200 years
7
3  10 s/yr
Physics 207: Lecture 2, Pg 9
Dimensional Analysis (reality check)

This is a very important tool to check your work
 Provides a reality check (if dimensional analysis fails
then there is no sense in putting in numbers)

Example
When 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
Physics 207: Lecture 2, Pg 12
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 13
Exercise 1
Dimensional Analysis
Which of the following formulas for F could be correct ?
Note: Force has dimensions of ML/T2
Velocity (n, dimension L / T)
A.
 F = mvR
Mass (m, dimension M)
B.
v
 F  m 
R
C.
2
mv
F 
R
2
Radius of the circle (R,
dimension L)
Physics 207: Lecture 2, Pg 14
Tracking changes in position: VECTORS

Position

Displacement (change in position)

Velocity (change in position with time)

Acceleration (change in velocity with time)

Jerk (change in acceleration with time)
Be careful, only vectors with identical units can be
added/subtracted
Physics 207: Lecture 2, Pg 18
Motion in One-Dimension (Kinematics)
Position

Position is usually measured and referenced to an origin:
10 meters
 At time= 0 seconds Pat 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
Displacement


One second later Pat is 15 meters to the right of the lamp
Displacement is just change in position

x = xf - xi
15 meters
10 meters
xi
Δx
xf
Pat
O
xf = xi + x
x = xf - xi = 5 meters to the right !
t = tf - ti = 1 second
Relating x to t yields velocity
Physics 207: Lecture 2, Pg 20
Average Velocity
Changes in position vs Changes in time
•
Average velocity = net distance covered per total
time, includes BOTH magnitude and direction
x(net displaceme nt )
vaverage velocity  
t ( total time )
x(5 m to the right )
vaverage velocity  
t (1 sec)
•
Pat’s average velocity was 5 m/s to the right
Physics 207: Lecture 2, Pg 21
Average Speed
Speed, s, is usually just the magnitude of velocity.
 The “how fast” without the direction.
 However:
Average speed references the total distance travelled

distance taken along path
saverage speed  
t ( total time )
•
Pat’s average speed was 5 m/s
Physics 207: Lecture 2, Pg 22
Exercise 2 Average Velocity
x (meters)
6
4
2
0
-2
1
2
3
4 t (seconds)
What is the magnitude of 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
Average Speed Exercise 4
What is the average speed over the first 4 seconds ?
0 m to -2 m to 0 m to 4 m  8 meters total
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 26
Instantaneous velocity
•
Instantaneous velocity, velocity at a given instant
x(displaceme nt ) dx
vvelocity   lim

t 0
t ( time )
dt
•
If a position vs. time curve then just the slope at a point
x
vinstantane ous
x f  xi
 lim
t 0 t  t
f
i
t
Physics 207: Lecture 2, Pg 27
Exercise 5
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 28
Exercise 5
Instantaneous Velocity
x (meters)
6
4
2
-2
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 29
Key point: position  velocity

(1) If the position x(t) is known as a function of time,
then we can find instantaneous or average velocity v
x  x(t )
dx
vx 
dt
t1
x   dt vx (t )
x
t
vx
t0


(2) “Area” under the v(t) curve yields the displacementt
A special case: If the velocity is a constant, then
t1
x  vx  dt  area of rectangle
t0
x(Δt)=v Δt + x0
Physics 207: Lecture 2, Pg 30
Acceleration


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
Now a “particle” with smoothly decreasing speed
v1
v0

v
v1

v0  v  v1
v2

v

v
 
v  v1  v0
v5
v4
v3

v

v

aavg

v

t
Physics 207: Lecture 2, Pg 35
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:

aavg
a

v

t
v0
v1
v2
v3
v4
v5
a
a
a
a
a
a
t
t
v
v(t)=v0 + a t
0
a t = area under curve = v (an integral)
t
Physics 207: Lecture 2, Pg 36
Instantaneous Acceleration

The instantaneous acceleration is the limit of the
average acceleration as ∆v/∆t approaches zero

Position, velocity & acceleration are all vectors and
must be kept separate (i.e., they cannot be added to
one another)
Physics 207: Lecture 2, Pg 37
Assignment

Reading for Tuesday’s class
» Finish Chapter 2 & all of 3 (vectors)
» HW 0 due soon, HW1 is available
Physics 207: Lecture 2, Pg 38