chapter 1
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Chapter 1
The Study of Motion
2
Units
We can classify almost all quantities in terms of the fundamental physical quantities: Length L Mass M Time T
For example: Speed has units L/T (miles per hour)
3
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)
4
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.
5
Units, cont’d
The back of your book provides numerous conversions. Here are some:
1 inch = 2.54 cm 1 m = 3.281 ft 1 mile = 5280 ft 1 km = 0.621 mi
6
Units, cont’d
We can use these to convert a compound unit:
mi 5280 ft 1 m 1 h 1min70
h 1 mi 3.281 ft 60min 60s
m31.2
s
7
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.
8
ExampleProblem 1.1
A yacht is 20 m long. Express this length in feet.
9
Example
A yacht is 20 m long. Express this length in feet.
3.281 ft20 m 20 3.281 ft
1 m65.62 ft
66 ft
ANSWER:
10
Example
How many liters are in a five gallon bucket? There are four quarts in a gallon.
11
Example
How many liters are in a five gallon bucket? There are four quarts in a gallon.
4 qt 0.95 L5 gal 5 4 0.95 L
1 gal 1 qt
19 L
ANSWER:
12
Metric prefixes
Sometimes a unit is too small or too big for a particular measurement.
To overcome this, we use a prefix.
13
Metric prefixes, cont’d
Power of 10 Prefix Symbol1015 peta P1012 tera T109 giga G106 mega M103 kilo k10-2 centi c10-3 milli m10-6 micro 10-9 nano n10-12 pico p10-15 femto f
14
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
15
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.
16
Frequency and period, cont’d
The symbols we use to represent frequency are period are: frequency: f period: T
They are related by
1 1period or
frequencyT
f
17
Frequency and period, cont’d
The standard unit of frequency is the Hertz (Hz). It is equivalent to 1 cycle per second.
18
ExampleExample 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?
19
ExampleExample 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?
10 cycles5 Hz.
2 sf
ANSWER:
20
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 distanceaverage speed
total elapsed time
21
Speed, cont’d
In mathematical notation,
So we can write speed as
final initial
final initial
total distance
total elapsed time
f i
f i
d d d d
t t t t
final initial
final initial
f i
f i
d dd dv
t t t t
d
t
22
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.
23
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
24
Speed, cont’d
We say that the distance is proportional to the elapsed time:
Using the speed gives us an equality, i.e., an equal sign, so we call v the proportionality constant.
d t
25
Speed, cont’d
Note that speed is relative. It depends upon what you are measuring your
speed against. Consider someone running on a ship.
26
Speed, cont’d
If you are on the boat, she is moving at
as seen on ship 8 mphv
27
Speed, cont’d
If you are on the dock, she is moving at
as seen on dock 8 mph 20 mph
28 mph
v
28
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?
29
ANSWER:
For the thunder:
For the flash:
Example
1600 m
345 m/s
4.6 s
soundsound
sound
dt
v
8
1600 m
3 10 m/s
0.0000053 s
lightlight
light
dt
v
30
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.”
31
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.
32
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.
33
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.
34
Vector addition, cont’d
Consider again someone running on a ship. If in the
same directions, the vectors add.
35
Vector addition, cont’d
Consider again someone running on a ship. If in the
opposite directions, the vectors subtract.
36
Vector addition, cont’d
What if the vectors are in different directions?
37
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.
38
Vector addition, cont’d
We can find the resulting magnitude of the Pythagorean theorem.
2 2 2
2 2
c a b
c a b
b
ac
39
Vector addition, cont’d
Let’s find the net speed of the bird? (Why didn’t I say net velocity?)
10
8
6
2 26 8 100
10
40
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.
41
Acceleration
Acceleration is the change in velocity divided by the elapsed time.
It measures the rate of change of velocity. Mathematically,
va
t
42
Acceleration, cont’d
The units are
In SI units, we might use m/s2. For cars, we might see mph/s.
2
/L T L La
T T T T
43
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.
44
Acceleration, cont’d
There is an important point to realize about acceleration:
It is the change in velocity.
45
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.
46
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.
47
ExampleExample 1.3
A car accelerates from 20 to 25 m/s in 4 seconds as it passes a truck. What is its acceleration?
48
ANSWER:
The problem gives us
The acceleration is:
ExampleExample 1.3
20 m/s
25 m/s
4 s
i
f
v
v
t
225 m/s 20 m/s1.25 m/s
4 s
f iv vva
t t
49
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.
ExampleExample 1.3
50
ExampleExample 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.
51
ANSWER:
The problem gives us
The acceleration is:
ExampleExample 1.3
9 m/s
0 m/s
5 s
i
f
v
v
t
20 m/s 9 m/s1.8 m/s
5 s
f iv vva
t t
52
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?
ExampleExample 1.3
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
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.
55
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.”
56
Centripetal acceleration, cont’d
For an object traveling with speed v on a circle of radius r , then its centripetal acceleration is
2va
r
57
Centripetal acceleration, cont’d
Note that the centripetal acceleration is: proportional to the speed-squared
inversely proportional to the radius
2a v
1a
r
58
ExampleExample 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.
59
ANSWER:
The problem gives us
The acceleration is:
ExampleExample 1.5
20 m
10 m/s
r
v
22
2 22
10 m/s
20 m
100 m /s5 m/s
20 m
va
r
60
ExampleProblem 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?
61
ANSWER:
The problem gives us
The acceleration is:
ExampleProblem 1.18
0.15 m
0.5 m/s
r
v
22
2 22
0.5 m/s
0.15 m
0.25 m /s1.7 m/s
0.15 m
va
r
62
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.
63
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.
64
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.
65
Simple types of motion— constant velocity, cont’d
The slope of the line gives us the speed.
66
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.
67
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
68
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:
2(9.8 m/s )
mph(22 )
s
v t
v t
69
Simple types of motion— constant acceleration, cont’d
If we plot speed versus time, the slope is the acceleration:
va
t
70
Simple types of motion— constant acceleration, cont’d
For an object starting from rest, v = 0, then the average speed is
12
0average speed
2
at
at
71
Simple types of motion— constant acceleration, cont’d
The distance is the average speed multiplied by the elapsed time:
12
212
d average speed t
at t
at
72
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.
2d t
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