Chapter 1
The Study of Motion
Units
 We can classify almost all quantities in terms
of the fundamental physical quantities:



Length
Mass
Time

L
M
T
For example:
 Speed has units L/T (miles per hour)
2
Units, cont’d
 SI (Système International) Units:
 MKS:




L = meters (m)
M = kilograms (kg)
T = seconds (s)
CGS:



L = centimeters (cm)
M = grams (g or gm)
T = seconds (s)
3
Units, cont’d
 British (or Imperial) Units:



L = feet (ft)
M = slugs or pound-mass (lbm)
T = seconds (s)
 We will use mostly SI but we need to know
how to convert back and forth.
4
Units, cont’d
 The back of your book provides numerous
conversions. Here are some:




1 inch
1m
1 mile
1 km
=
=
=
=
2.54 cm
3.281 ft
5280 ft
0.621 mi
5
Units, cont’d
 We can use these to convert a compound
unit:
mi 5280 ft
1m
1h
1min
70 



h
1 mi
3.281 ft 60 min 60s
m
 31.2
s
6
Converting units
 Look at your original units.
 Determine the units you want to have.
 Find the conversion you need.
 Write the conversion as a fraction that
replaces the original unit with the new unit.
7
Example
Problem 1.1
A yacht is 20 m long. Express this length in feet.
8
Example
A yacht is 20 m long. Express this length in feet.
ANSWER:
3.281 ft
20 m 
 20  3.281 ft
1m
 65.62 ft
 66 ft
9
Example
How many liters are in a five gallon bucket?
There are four quarts in a gallon.
10
Example
How many liters are in a five gallon bucket?
There are four quarts in a gallon.
ANSWER:
4 qt 0.95 L
5 gal 

 5  4  0.95 L
1 gal 1 qt
 19 L
11
Metric prefixes
 Sometimes a unit is too small or too big for a
particular measurement.
 To overcome this, we use a prefix.
12
Metric prefixes, cont’d
Power of 10
1015
1012
109
106
103
10-2
10-3
10-6
10-9
10-12
10-15
Prefix
peta
tera
giga
mega
kilo
centi
milli
micro
nano
pico
femto
Symbol
P
T
G
M
k
c
m
m
n
p
f
13
Metric prefixes, cont’d
 Some examples:



1 centimeter = 10-2 meters = 0.01 m
1 millimeter = 10-3 meters = 0.001 m
1 kilogram = 103 grams = 1,000 g
14
Frequency and period
 We define frequency as the number of
events per a given amount of time.
 When an event occurs repeatedly, we say
that the event is periodic.
 The amount of time between events is the
period.
15
Frequency and period, cont’d
 The symbols we use to represent frequency
are period are:


frequency: f
period: T
 They are related by
1
1
period 
or T 
frequency
f
16
Frequency and period, cont’d
 The standard unit of frequency is the Hertz
(Hz).

It is equivalent to 1 cycle per second.
17
Example
Example 1.1
A mechanical stopwatch uses a balance wheel
that rotates back and forth 10 times in 2
seconds. What is the frequency of the
balance wheel?
18
Example
Example 1.1
A mechanical stopwatch uses a balance wheel
that rotates back and forth 10 times in 2
seconds. What is the frequency of the
balance wheel?
ANSWER:
10 cycles
f 
 5 Hz.
2s
19
Speed
 Speed is the rate of change of distance from
a reference point.
 It is the rate of movement.
 It equals the distance something travels
divided by the elapsed time.
total distance
average speed 
total elapsed time
20
Speed, cont’d
 In mathematical notation,
total distance  dfinal  dinitial  d f  di
total elapsed time  tfinal  tinitial  t f  ti
 So we can write speed as
d final  dinitial d f  di
v

tfinal  tinitial
t f  ti
d

t
21
Speed, cont’d
 The symbol  is the Greek letter delta and
represents the change in.
 As the time interval becomes shorter and
shorter, we approach the instantaneous
speed.
22
Speed, cont’d
 If we know the average speed and how long
something travels at that speed, we can find
the distance it travels:
d  vt
23
Speed, cont’d
 We say that the distance is proportional to
the elapsed time:
d t
 Using the speed gives us an equality, i.e., an
equal sign, so we call v the proportionality
constant.
24
Speed, cont’d
 Note that speed is relative.

It depends upon what you are measuring your
speed against.
 Consider someone running on a ship.
25
Speed, cont’d
 If you are on the boat, she is moving at
vas seen on ship  8 mph
26
Speed, cont’d
 If you are on the dock, she is moving at
vas seen on dock  8 mph  20 mph
 28 mph
27
Example
When lightning strikes, you see the flash almost
immediately but the thunder typically lags
behind. The speed of light is 3 × 108 m/s and
the speed of sound is about 345 m/s. If the
lightning flash is one mile away, how long
does it take the light and sound to reach you?
28
Example
ANSWER:
For the thunder:
tsound
d sound 1600 m


vsound 345 m/s
 4.6 s
For the flash:
tlight 
dlight
vlight
1600 m

3 108 m/s
 0.0000053 s
29
Velocity
 Velocity is the speed in a particular direction.
 It tells us not only “how fast” (like speed) but
also how fast in “what direction.”
30
Velocity, cont’d
 In common language, we don’t distinguish
between the two.

This sets you up for confusion in a physics
class.
 During a weather report, you might be given
the wind-speed is 15 mph from the west.
31
Velocity, cont’d
 The speed of the wind is 15 mph.
 The wind is blowing in a direction from the
west to the east.
 So you are actually given the wind velocity.
32
Vector addition
 Quantities that convey a magnitude and a
direction, like velocity, are called vectors.
 We represent vectors
by an arrow.

The length indicates
the magnitude.
33
Vector addition, cont’d
 Consider again someone running on a ship.

If in the
same
directions,
the vectors
add.
34
Vector addition, cont’d
 Consider again someone running on a ship.

If in the
opposite
directions,
the vectors
subtract.
35
Vector addition, cont’d
 What if the vectors are in different directions?
36
Vector addition, cont’d
 The resulting velocity of the bird (from the
bird’s velocity and the wind) is a combination
of the magnitude and direction of each
velocity.
37
Vector addition, cont’d
 We can find the resulting magnitude of the
Pythagorean theorem.
c  a b
2
2
2
c  a b
2
a
c
2
b
38
Vector addition, cont’d
 Let’s find the net speed of the bird?
(Why didn’t I say net velocity?)
6
6  8  100
 10
2
2
8
10
39
Vector addition, cont’d
 Here are more examples, illustrating that
even if the bird flies with the same velocity,
the effect of the wind can be
constructive
or destructive.
40
Acceleration
 Acceleration is the change in velocity
divided by the elapsed time.
 It measures the rate of change of velocity.
 Mathematically,
v
a
t
41
Acceleration, cont’d
 The units are
L /T
L
L
a

 2
T
T T T
 In SI units, we might use m/s2.
 For cars, we might see mph/s.
42
Acceleration, cont’d
 A common way to express acceleration is in
terms of g’s.
 One g is the acceleration an object
experiences as it falls near the Earth’s
surface: g = 9.8 m/s2.

So if you experience 2g during a collision,
your acceleration was 19.6 m/s2.
43
Acceleration, cont’d
 There is an important point to realize about
acceleration:
It is the change in velocity.
44
Acceleration, cont’d
 Since velocity is speed and direction, there
are three ways it can change:



change in speed,
change in direction, or
change in both speed & direction.
 The change in direction is an important case
often misunderstood.
45
Acceleration, cont’d
 If you drive through a curve with the cruise
control set to 65 mph, you are accelerating.


Not because your speed changes.
But because your direction is changing.


There must be an acceleration because items on
your dash go sliding around.
More on this in chapter 2.
46
Example
Example 1.3
A car accelerates from 20 to 25 m/s in
4 seconds as it passes a truck. What is its
acceleration?
47
Example
Example 1.3
ANSWER:
The problem gives us
vi  20 m/s
v f  25 m/s
t  4 s
The acceleration is:
v v f  vi
a

t
t
25 m/s  20 m/s

 1.25 m/s 2
4s
48
Example
Example 1.3
CHECK:
Does this make sense?
The car needs to increase its speed
5 m/s in 4 seconds.
If it increased 1 m/s every second, it would
only reach 24 m/s.
So we should expect an answer slightly more
than 1 m/s every second.
49
Example
Example 1.4
After a race, a runner takes 5 seconds to come
to a stop from a speed of 9 m/s. Find her
acceleration.
50
Example
Example 1.3
ANSWER:
The problem gives us
vi  9 m/s
v f  0 m/s
t  5 s
The acceleration is:
v v f  vi
a

t
t
0 m/s  9 m/s

 1.8 m/s 2
5s
51
Example
Example 1.3
CHECK:
Does this make sense?
If she was traveling at 10 m/s, reducing her
speed 2 m/s every second would stop her in
5 seconds.
What’s up with the minus sign?
52
Centripetal acceleration
 Remember that acceleration can result from a
change in the velocity’s direction.
 Imagine a car rounding a curve.
 The car’s velocity must keep changing toward
the center of the curve in order to stay on the
road.
53
Centripetal acceleration
 Remember that acceleration can result from
a change in the velocity’s direction.
 Imagine a car rounding a curve.
 The car’s velocity
must keep
changing toward
the center of the
curve in order to
stay on the road.
54
Centripetal acceleration, cont’d
 So there is an acceleration toward the center
of the curve.
 Centripetal acceleration is the acceleration
associated with an object moving in a circular
path.

Centripetal means “center-seeking.”
55
Centripetal acceleration, cont’d
 For an object traveling with speed v on a
circle of radius r , then its centripetal
acceleration is
2
v
a
r
56
Centripetal acceleration, cont’d
 Note that the centripetal acceleration is:

proportional to the speed-squared
av

2
inversely proportional to the radius
1
a
r
57
Example
Example 1.5
Let’s estimate the acceleration of a car as it
goes around a curve. The radius of a
segment of a typical cloverleaf is 20 meters,
and a car might take the curve with a
constant
speed of
10 m/s.
58
Example
Example 1.5
ANSWER:
The problem gives us
r  20 m
v  10 m/s
The acceleration is:
10 m/s 

v
a 
r
20 m
100 m 2 /s 2

 5 m/s 2
20 m
2
2
59
Example
Problem 1.18
An insect sits on the edge of a spinning record
that has a radius of 0.15 m. The insect’s
speed is about 0.5 m/s when the record is
turning at
33-1/3 rpm. What is the insect’s acceleration?
60
Example
Problem 1.18
ANSWER:
The problem gives us
r  0.15 m
v  0.5 m/s
The acceleration is:
0.5 m/s 

v
a 
r
0.15 m
0.25 m 2 /s 2

 1.7 m/s 2
0.15 m
2
2
61
Simple types of motion
— zero velocity
 The simplest type of motion is obviously no
motion.
 The object has no velocity.
 So it never moves.
 The position of the object, relative to some
reference, is constant.
62
Simple types of motion
— constant velocity
 The next simplest type of motion is uniform
motion.

In physics, uniform means constant.
 The object’s velocity does not change.
 So its position, relative to some reference, is
proportional to time.
63
Simple types of motion
— constant velocity, cont’d
 If we plot the object’s distance versus time,
we get this graph.

Notice that if we double the time interval, then
we double the object’s distance.
64
Simple types of motion
— constant velocity, cont’d
 The slope of the line gives us the speed.
65
Simple types of motion
— constant velocity, cont’d
 If an object moves faster, then the line has a
larger speed.
 So the graph has
a steeper slope.
66
Simple types of motion
— constant acceleration
 The next type of motion is uniform
acceleration in a straight line.
 The acceleration does not change.
 So the object’s speed is proportional to the
elapsed time.
speed  acceleration  time
v  at
67
Simple types of motion
— constant acceleration, cont’d
 A common example is free fall.

Free fall means gravity is the only thing
changing an object’s motion.
 The speed is:
v  (9.8 m/s ) t
mph
v  (22
)t
s
2
68
Simple types of motion
— constant acceleration, cont’d
 If we plot speed versus time, the slope is the
acceleration:
v
a
t
69
Simple types of motion
— constant acceleration, cont’d
 For an object starting from rest, v = 0, then
the average speed is
0  at
average speed 
2
 12 at
70
Simple types of motion
— constant acceleration, cont’d
 The distance is the average speed multiplied
by the elapsed time:
d  average speed  t
 at  t
1
2
 at
1
2
2
71
Simple types of motion
— constant acceleration, cont’d
 If we graph the distance
versus time, the curve is
not a straight line.

The distance is
proportional to the
square of the time.
dt
2
72