Chapter 1 Notes

Report
Chapter 1
Introduction and
Mathematical Concepts
Objectives
• To express real numbers in scientific
notation and solve using a calculator
• To determine the number of sig figs in a
given number
• To solve equations using only variables
Scientific Notation
• Scientific notation: useful for small
and large #’s
• Use a coefficient raised to the power
of 10
• Coefficient must be between 1 and 10
• Use a power of 10 depending on how
many times you moved the decimal
place
Scientific Notation
• Example:
–10,000,000 becomes 1 x107
–0.0000025 becomes 2.5 x10-6
• Greater than 1= positive power
• Less than 1= negative power
Scientific Notation Examples
• Express in scientific Notation
• .00000000001874
• 1.874 x 10-11
Scientific Notation Examples
• Express in scientific Notation
• 200,000,000,000
• 2 x 1011
Scientific Notation Examples
• Express in scientific Notation
• .00076321
• 7.6321 x 10-4
Scientific Notation Examples
• Express in scientific Notation
• 984.73
• 9.8473 x 102
• Explanation: Why is this example
odd?
Scientific Notation Examples
• Express as real numbers
• 1.34 x 105
• 134,000
Scientific Notation Examples
• Express as real numbers
• 6.0 x 10-3
• .0060
Scientific Notation Examples
• Express as real numbers
• 5.223 x 108
• 522,300,000
Objectives
• To express real numbers in scientific
notation and solve using a calculator
• To determine the number of sig figs in a
given number
• To solve equations using only variables
Scientific Notation and
Calculators
• When using a calculator DO NOT type in
x10 meaning DO NOT Push the buttons
[x], [1] & [0].
• Depending on your calculator you will use
the [EE], [Exp] or [x10n] button when a
number is in scientific notation
Scientific Notation
• What is the answer to…
• 642 x (4.0 x 10-5)
• 2.6 x 10-2 or .02568
Scientific Notation
• What is the answer to…
• 1.7 x 105 / 3.88 x 107
• 4.4 x 10-3 or .0044
Scientific Notation
• What is the answer to…
• 2.9 x 10-5 x (8.1 x 102)
• 2.3 x 10-2 or .023
Scientific Notation
• What is the answer to…
• 6.02 x 1023 / (7.2 x 108)
• 8.4 x 1014
Scientific Notation
• What is the answer to…
• 5.40 x10-18 / 769
• 7.02 x 10-21
Scientific Notation
• What is the answer to…
• (1.0 x 107) x (1.0 x 10-6)
• 10
Scientific Notation Examples
• Express as real numbers
• 1.59 x 10-2
• .0159
Scientific Notation Sample Problems
• Put the following numbers in
scientific notation:
–4500000
–0.00234
–168000000000
–0.00000000036
Scientific Notation Sample Problems
• Answers:
–4.5 x106
–2.34 x10-3
–1.68 x1011
–3.6 x10-10
Objectives
• To express real numbers in scientific
notation and solve using a calculator
• To determine the number of sig figs in a
given number
• To solve equations using only variables
Significant Figures
• Significant figures: (also called significant
digits) of a number are those digits that
carry meaning contributing to its precision.
This includes all digits except:
• leading and trailing zeros which are merely
placeholders to indicate the scale of the
number.
• spurious digits introduced
• which instrument gives us more sigfigs?
• that’s the one you want to use (but it probably
costs a lot more!)
Rules for Significant Figures
• 1) The #’s 1-9 are always significant
• Examples:
–2467
–4 sig. figs.
–2.344678,
–7 sig. figs.
Rules for Significant Figures
• 2) A zero between 2 significant
figures is always significant
• Examples:
– 23045,
– 5 sig. figs.
– 450000001,
– 9 sig. figs. Because all zeros are
between significant figures
Rules for Significant Figures
• 3) Place holders are not significant,
so a zero in a decimal before a
significant figure is not counted
• Example:
– 0.023,
– 2 sig. figs. Zeroes come before a
significant figure
– 0.00000004,
– 1 sig. fig., all zeroes come before a
significant figure
Rules for Significant Figures
• 4) Zeroes after a significant figure and
after a decimal are counted.
• Example:
– 0.00230,
– 3 sig. figs. The zero after the 3 is after a
significant number and a decimal so it is
counted
– 23.34000,
– 7 sig. figs.
Rules for Significant Figures
• 5) Zeroes after a significant figure
when there is no decimal are place
holders so they are not counted.
• Example:
– 1200
– 2 sig. figs. Zeros are place holders
– 145670,
– 5 sig. figs., last decimal is a place
holder
Rules for Significant Figures
• 6) If you count to have an exact
quantity, you can use unlimited
significant figures.
• Example:
– You counted 18 sheep, you could write
the number as 18.0000000 and have 9
significant figures
– You can use as many significant figures
as you want because you know you
have exactly 18 sheep
Significant Figure Sample Problem
• How many significant
figures do each of
• Answers:
these have?
– 15.008
– 146000
– 0.00025760
– 2357
– 6.0 x 105
– 15.008 = 5
– 146000 = 3
– 0.00025760 = 5
– 2357 = 4
– 6.0 x 105 = 2
Objectives
• To express real numbers in scientific
notation and solve using a calculator
• To determine the number of sig figs in a
given number
• To solve equations using only variables
Solving Equations Using
Variables
• 2 Rules: The variable you are solving for
must…
1. Be by itself on one side of the equal sign
2. In the numerator
• Look to worksheet for examples
Objectives
• We will be able to convert between
different metric units given a conversion
chart
• We will be able to convert between units
using dimensional analysis
• We will be able to solve right triangle
problems using sine, cosine, and tangent
1.2 Units
SI units
Le Système International d’Unités
(What the rest of the world uses
Length: meter (m)
Mass: kilogram (kg)
Time: second (s)
1.2 Units
The Standard
Platinum-Iridium Meter
Bar kept at 0°C
1.2 Units
The standard
platinum-iridium
kilogram
1.3 The Role of Units in Problem Solving
REMEMBER THIS!
The Great Mighty Kids
Have Dropped Over
Dead Converting
Metrics Many Nights
Past Friday
1.2.8. How many meters are there in 12.5 kilometers?
a) 1.25m
b) 125m
c) 1250m
d) 12 500m
e) 125 000m
1.3 The Role of Units in Problem Solving
Reasoning Strategy: Converting Between Units
1. In all calculations, write down the units explicitly.
2. Treat all units as algebraic quantities. When
identical units are divided, they are eliminated
algebraically.
Objectives
• We will be able to convert between
different metric units given a conversion
chart
• We will be able to convert between units
using dimensional analysis
• We will be able to solve right triangle
problems using sine, cosine, and tangent
1.3 The Role of Units in Problem Solving
Example 1 The World’s Highest Waterfall
•The highest waterfall in the world is Angel Falls in Venezuela,
with a total drop of 979.0 m. Express this drop in feet.
•Since 3.281 feet = 1 meter, it follows that
Length
 3 . 281 feet 
 979 . 0 meters 
  3212 feet
 1 meter 
1.3 The Role of Units in Problem Solving
Example 2 Interstate Speed Limit
Express the speed limit of 65 miles/hour in terms
of meters/second.
Use 5280 feet = 1 mile and 3600 seconds = 1
hour and 3.281 feet = 1 meter.
miles 

Speed   65
 1 1  
hour 

miles   5280 feet   1 hour 
feet

65

95




hour
mile
3600
s
second




feet 

Speed   95
 1  
second 

feet   1 meter 
meters

 95

  29
second   3.281 feet 
second

Objectives
• We will be able to convert between different
metric units given a conversion chart
• We will be able to convert between units using
dimensional analysis
• We will be able to solve right triangle
problems using sine, cosine, and tangent
1.4 Trigonometry
The sides of a right triangle
1.4 Trigonometry
Sine, Cosine and Tangent
sin  
ho
h
Abbrev. SOH CAH TOA
cos  
ha
h
tan  
ho
ha
1.4.3. Referring to the triangle with sides labeled A, B,
and C as shown, which of the following ratios is equal
to the sine of the angle ?
a)
A
B
b)
A
C
c)
B
C
d)
B
A
e)
C
B
1.4.4. Referring to the triangle with sides labeled A, B,
and C as shown, which of the following ratios is equal
to the tangent of the angle  ?
a)
A
B
b)
A
C
c)
B
C
d)
B
A
e)
C
B
1.4 Trigonometry
Trig Example: Solve for ho
tan  
ho
ha
tan 50 

ho
67 . 2 m
h o  tan 50 67 . 2 m   80 . 0 m

1.4 Trigonometry
When solving for the
angle, you must use the
inverse trig function
  sin
1
 ho 


 h 
  cos
1
  tan
1
 ha 


 h 
 ho 


h 
 a 
1.4 Trigonometry
Solve for the angle
  tan
1
 ho 


h 
 a 
  tan
1
 2 . 25 m 


  9 . 13
 14 . 0 m 
1.4 Trigonometry
Pythagorean theorem: When you know 2 sides
of a right triangle and you are trying to solve
for the 3rd
2
2
2
h  ho  h a
Objectives
• We will be able to identify vectors and
scalars when given a quantity
• We will be able to solve problems that
involve adding or subtracting vectors that
are parallel, perpendicular and skewed
1.5 Scalars and Vectors
A scalar quantity is one that can be described
by a single number:
temperature, speed, mass
A vector quantity deals inherently with both
magnitude and direction:
velocity, force, displacement
1.5.2. Which one of the following quantities is a
vector quantity?
a) the age of the pyramids in Egypt
b) the mass of a watermelon
c) the sun's pull on the earth
d) the number of people on board an airplane
e) the temperature of molten lava
1.5.4. Which one of the following situations involves a
vector quantity?
a) The mass of the Martian soil probe was 250 kg.
b) The overnight low temperature in Toronto was 4.0 C.
c) The volume of the soft drink can is 0.360 liters.
d) The velocity of the rocket was 325 m/s, due east.
e) The light took approximately 500 s to travel from the
sun to the earth.
1.5 Scalars and Vectors
Arrows are used to represent vectors. The
direction of the arrow gives the direction of
the vector.
By convention, the length of a vector
arrow is proportional to the magnitude
of the vector.
Examples
4 lb
8 lb
Objectives
• We will be able to identify vectors and
scalars when given a quantity
• We will be able to solve problems that
involve adding or subtracting vectors that
are parallel, perpendicular and skewed
1.6 Vector Addition and Subtraction
Often it is necessary to add or subtract one
vector with another.
We will use the tail to head method
The combined vector is called the RESULTANT
1.6 Vector Addition and Subtraction
If the vectors are facing
the SAME DIRECTION
we can simply add their
magnitudes
3m
5m
8m
1.6 Vector Addition and Subtraction
Using the tail to head method, the
resultant goes from the TAIL of the 1st
vector to the HEAD of the 2nd vector
1.6.3. Consider the two vectors represented in the drawing. Which of
the following options is the correct way to add graphically vectors
a and b ?
1.6 Vector Addition and Subtraction
If 2 vectors are
PREPENDICULAR
then we will use the
PYTHAGOREAN
THEOREM to solve
for the 3rd side.
Solve for the 3rd
side and the angle
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
R   2 . 00 m   6 . 00 m 
2
2
R 
 2 . 00
2
m   6 . 00 m   6 . 32 m
2
2
R
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
tan   2 . 00 6 . 00
  tan
1
2 .00
6 . 00   18 . 4

6.32 m
2.00 m

6.00 m
1.6 Vector Addition and Subtraction
When a vector is
multiplied by -1, the
magnitude of the
vector remains the
same, but the
direction of the vector
is reversed.
1.6 Vector Addition and Subtraction

B


AB

A

A

B
 
AB
1.6.4. Consider the two vectors represented in the drawing. Which of
the following options is the correct way to subtract graphically
vectors a and b ?
1.7 The Components of a Vector


x and y are called the x vector component

and the y vector component of r .
1.7 The Components of a Vector

of A are two perpendicu
The vector components
lar


vectors A x and A y that are parallel to the x and y axes,



and add together vectoriall y so that A  A x  A y .
1.7 The Components of a Vector
Example
A displacement vector has a magnitude of 175 m and points at
an angle of 50.0 degrees relative to the x axis. Find the x and y
components of this vector.
sin   y r
y  r sin   175 m sin 50 . 0

  134
m

  112
m
cos   x r
x  r cos   175 m cos 50 . 0

r  112 m xˆ  134 m yˆ
1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25
relative to the x axis. Which of the following choices indicates the
horizontal and vertical components of vector r ?
a)
rx
+22 km/h
ry
+66 km/h
b)
+39 km/h
+79 km/h
c)
+79 km/h
+39 km/h
d)
+66 km/h
+22 km/h
e)
+72 km/h
+48 km/h
1.8 Addition of Vectors by Means of Components
Notice in this example the 2 vectors being
added are not perpendicular. We have to add
BY COMPONENTS



C  AB

A  A x xˆ  A y yˆ

B  B x xˆ  B y yˆ
1.8 Addition of Vectors by Means of Components

C  A x xˆ  A y yˆ  B x xˆ  B y yˆ
  A x  B x xˆ   A y  B y yˆ
C x  Ax  B x
C y  Ay  B y
1.8.1. V ector A has scalar com ponents Ax = 35 m /s and
Ay = 15 m /s. V ector B has scalar com ponen ts B x =  22 m /s
and B y = 18 m /s. D eterm ine the scalar com ponents of vector
C = A  B.
a)
Cx
13 m/s
Cy
3 m/s
b)
57 m/s
33 m/s
c)
13 m/s
33 m/s
d)
57 m/s
3 m/s
e)
57 m/s
3 m/s

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