Basic Math for Physics

AP Physics Rapid Learning Series - 02
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Basic Math foras c at o
Physics
Physics Rapid Learning Series
www.RapidLearningCenter.com/
© Rapid Learning Inc. All rights reserved.
Wayne Huang, Ph.D.
Keith Duda, M.Ed.
Peddi Prasad, Ph.D.
Gary Zhou, Ph.D.
Michelle Wedemeyer, Ph.D.
Sarah Hedges, Ph.D.

Learning Objectives
Using algebra to solve for a
i bl
By completing this tutorial, you will learn math
skills commonly used in physics:
variable
Performing calculations
with significant figures
Using scientific notation
Performing calculations
with exponents & scientific
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with exponents & scientific
notation
Trigonometry
Using the quadratic formula
Calculator tips
Concept Map
experimental science
Physics is an
experimental science calculator tips
Follow
calculator tips
Requires
Previous content
New content
ExperimentationExperimentation Conclusions
Results &
Conclusions
Data taken with
correct # of sig figs
Data taken with
correct # of sig figs
CalculationsCalculations
Follow rules
with sig figs
Follow rules
for calculations
with sig figs
Leads to
Often requires
During which
When doing,
Be sure to
May be written in
Depend on
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QuadraticQuadratic
Formula
TrigonometryTrigonometryAlgebra
Rules for
exponents
Rules for
working with
exponents
Use scientificUse scientific
notation
Might use
When using, follow

Algebra
Algebra is often used when
sol ing ph sics problems
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solving physics problems.
Algebra Rules
If a number is
being … to an
Then … that
number on both
Undo by doing the opposite to both sides:
Example
unknown
variable
sides
Added Subtract
Example
5 = x + 2
-2 -2
5-2 = x
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Subtracted Add
3 = x – 6
+6 +6
3+6 = x

Algebra Rules-2
If a number is
being … to an
Then … that
number on both
Undo by doing the opposite to both sides:
Example
2 = 4x
4 4
2÷4 = x
unknown
variable
sides
Multiplied Divide
Example
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Divided Multiply
2* 6 = x *2
2
2*6 = x
Example: Velocity
Velocity problems require algebraic work:
displacement(d)
Velocity
time(t)
=
Example: Solve for distance
The unknown variable (d) is being divided by 25 s
time(t)
25s
d
m/s52 = * 25 s25 s *
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The unknown variable (d) is being divided by 25 s.
Multiply both sides by 25 s
dm/s52*25s =
d1300m =

Example: Force
Net force problems also require algebraic work:
frictionappliednet FFF +=
frictionF52N25N +=
Example: Solve for force of friction
The unknown variable (Ff i ti ) is being
-52 N-52 N
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The unknown variable (Ffriction) is being
added to 52 N
Subtract 52 N from both sides
frictionF52N25N =−
frictionF27N =−
Example: Gravitational Force
When solving for a variable on the bottom, you need to
move it to the top first:
Example: Solve for d
( )22118 g)(50kg)(50k
/kgNm106 6N101 7 −−
×=×
Move d to top
( ) 2
d
/kgNm106.6N101.7 ×=×
( ) 2
22118
d
g)(50kg)(50k
/kgNm106.6N101.7 −−
×=×d2 * * d2
( ) ( ) g)(50kg)(50k/kgNm106.6N101.7d 221182 −−
×=×
Solve for d
10/67
( ) ( )
1.7x10-8 N 1.7x10-8 N
N)10(1.7
kg))(50kg)(50/kgNm10(6.6
d 8
2211
−
−
×
×
=
3.11md =

Calculations &
Significant Figures
This section describes the role
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of significant figures in
calculations.
Calculations, Significant Figures Usage
Data is taken very carefully to the correct number of
significant figuressignificant figures.
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When performing calculations, you cannot end up
with greater precision than what you actually
measured in the lab!

Always Round Last
Always complete all calculations first…
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And then round as a final step.
Rounding: Adding & Subtracting
Adding & subtracting:
Perform the calculation1
Example: Perform the following operation:
10.027 g
1 5 g
3 decimal places
1 decimal place
Determine the least # of decimal places in the problem
Round the answer to that many decimal places.
2
3
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- 1.5 g
8.527 g
1 decimal place
8.5 g
1
Least = 1
2
3

Rounding: Multiplying & Dividing
Multiplying & Dividing:
Perform the calculation1
Example: Perform the following operation:
Determine the least # of significant figures in the
problem
Round the answer to that many significant figures.
2
3
1
10.027 m
6 68467 m/s
5 sig figures
=
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6.68 m/sLeast = 3
2
3
1.50 s
6.68467 m/s
3 sig figures
=
Scientific Notation
Scientific notation is often used
th h t th t d f h i
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throughout the study of physics.

Definition – Scientific Notation
Scientific Notation - A short-hand
method of writing very large or very
ll bsmall numbers.
Scientific Notation uses powers of
10.
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Writing in Scientific Notation
Rules for writing scientific notation:
Decimal is moved to follow the 1st non-zero number
Number of times the decimal is moved = power of 10
1
2
Examples: Write in scientific notation:
1027500.456 g
1
“Big” (>1) numbers positive powers
“Small” (<1) numbers negative powers
2
3
123456
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1.027500456 g
The decimal point was moved 6 times 1.027500456
The original number was “big” so it will be a +6 power
1.027500456 x 106 g
1
2
3
123456

Writing in Scientific Notation 2
Rules for writing scientific notation:
Decimal is moved to follow the 1st non-zero number
1
2
Examples: Write in scientific notation:
0.0007543 g
1
“Big” (>1) numbers positive powers
“Small” (<1) numbers negative powers
2
3
1234
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7.543 g
The decimal point was moved 4 times 0007.543
The original number was “small” so it will be a -4 power
7.543 x 10-4 g
1
2
3
1234
Sig Figures & Scientific Notation
Scientific Notation can also be used to write a
specific number of significant figures.
Example: Your calculator gives you: 10200 g
And you need to write the result with 4 significant figures
10200 g 3 significant figures
10200. g 5 significant figures
4321
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1.020 X 104 g 4 significant figures!
Use scientific notation
4321

Reading Scientific Notation - 1
Rules for reading scientific notation:
Power of 10 = number of times to move the decimal
P iti “Bi ” ( 1) b
1
Examples: Write out the following number
1
Positive powers “Big” (>1) numbers
Negative powers “Small” (<1) numbers
2
3.25 X 10-6 g00000
123456
.
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1
2
The decimal point will move 6 times
The power is negative, so we’ll make the number
“small”—move the decimal forward (left)
0.00000325 g
123456
Reading Scientific Notation - 2
Rules for reading scientific notation:
Power of 10 = number of times to move the decimal
P iti “Bi ” ( 1) b
1
Examples: Write out the following number
1
Positive powers “Big” (>1) numbers
Negative powers “Small” (<1) powers
2
7.2004 X 103 mL.
123
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1
2
The decimal point will move 3 times
The power is positive, so we’ll make the number
“big”—move the decimal point right (backwards)
7200.4 mL
123

Exponents
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Exponent Use
Exponents are used in unit
conversions and when using
scientific notation.
There are several
rules that govern
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g
calculations with
exponents.

Exponent Rule #1
“Rule of 1”
A b t th f 1 th i i l b1
Example:
Any number to the power of 1 = the original number1
251 = 25
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Exponent Rule #2
“Rule of 0”
A b t th f 0 12
Example:
Any number to the power of 0 = 12
160 = 1
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Exponent Rule #3
“Product rule”
Wh lti l i dd th t3
Example:
When multiplying, add the exponents.3
112*114
= 11(2+4)
This rule does not apply when the bases
= 11(6)
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pp y
are different!
Example:
112*124 (11*12)(2+4)=
Exponent Rule #4
“Quotient rule”
Wh di idi bt t th t4
Example:
When dividing, subtract the exponents.4
= 15(8-3)
This rule does not apply when the bases
= 15(5)
158
153
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pp y
are different!
Example:
158
123 (15
12)
(8-3)
=

Exponent Rule #5
“Power rule”
When there is an exponent on an exponent multiply5
Example:
When there is an exponent on an exponent, multiply
the two exponents together.
5
(82)3
= 8(2*3)
= 8(6)
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(8 ) = 8( )
= 8( )
Exponent Rule #6
“Negative rule”
When a number with a negative exponent is placed on6
Example:
When a number with a negative exponent is placed on
the opposite side of the fraction, the exponent is
made positive.
6
9-2
=
1
92
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9 = 92

Summary of Exponent Rules
A b t th f 0 1
Any number to the power of 1 = the original number.
251 = 25
1
Exponent on an exponent: multiply the two exponents5
Dividing: subtract the exponents. 32÷38 = 3-6
4
Multiplying: add the exponents. 32*38 = 310
3
Any number to the power of 0 = 1 250 = 12
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Negative exponent: place on the opposite side of the
fraction and the exponent is made positive 3-2 = 1/32
6
together. (32)3 = 36
5
Calculations with Scientific Notation
There are severale e a e se e a
similar rules when
working with
scientific notation
calculations.
Scientific
Notation
6.02x1023
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Sci. Notation Calculation Rule #1
Addition
Add th b d k th f 101
Example:
Add the numbers and keep the same power of 101
6x104 + 2x104
= (6+2)x104
This rule does not apply when the powers
= 8x104
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pp y p
of 10 are different!
Example:
6x103 + 6x106 (6+6)x10(3+6)=
Subtraction
S bt t th b d k th f 102
Example:
Subtract the numbers and keep the same power of 102
3x102 - 1x102
= (3-1)x102
This rule does not apply when the powers
= 2x102
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pp y p
of 10 are different!
Example:
3x102 - 6x106 (3-6)x10(2-6)=

Multiplication
M lti l th b d dd th f 103
Example:
Multiply the numbers and add the powers of 10.3
3x104 * 2x106
= (3*2)x10(4+6)
= 6x1010
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Division
Di id th b d bt t th f 104
Example:
Divide the numbers and subtract the powers of 10.4
8x102 ÷ 2x106
= (8 ÷ 2)x10(2-6)
= 4x10-4
36/67

Taking a scientific notation number to a power
T k th b t th d lti l th5
Example:
Take the number to the power and multiply the power
by the power of 10.
5
(3x102)3
= (3)3x10(2*3)
= 27x106
37/67
Taking a root of a scientific notation number
T k th t f th b d di id th f6
Example:
Take the root of the number and divide the power of
10 by the root.
6
= =√3x102 √3x10(2/2)
√3x10(1)
38/67

Summary of Sci. Notation Rules
Subtraction: Subtract the numbers and keep the same2
Addition: Add the numbers and keep the same power of
10 2X103 + 3X103 = 5X103
1
Division: Divide the numbers and subtract the powers of
10 2X104 / 3X103 = 0.67X101
4
Multiplication: Multiply the numbers and add the powers
of 10 2X106 * 3X103 = 6X109
3
Subtraction: Subtract the numbers and keep the same
power of 10 2X103 - 3X103 = -1X103
2
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Roots: Take the root of the number and divide the power
of 10 by the root √(3x102)= √3 X 101
6
Powers: Take the number to the power and multiply the
power by the power of 10 (2X103)3 = 8X109
5
Geometry &
Trigonometry
Both aspects of math are
frequently used in physics
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frequently used in physics.

Definition – Trigononmetric Functions
Trigonometric Functions – Functions
of an angle that give ratios of theof an angle that give ratios of the
sides of a triangle within that right
triangle.
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Trigonometric Functions
The three common trig functions work with right
triangles.
itTrig functions use one of the
hypotenuse
opposite
sinθ =
hypotenuse
adjacent
cosθ =
Trig functions use one of the
non-90° angles and discuss the
sides of the triangle as they
relate to that angle
tetoθ
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adjacent
opposite
tanθ =
Right angle θ“Theta”—angle in
question
Opposit
Adjacent to θ

Remembering the Functions
S in
Opposite Make sure you
Hypotenuse
C os
Adjacent
Hypotenuse
Make sure you
pronounce it with
a long “O” in
“SOH” so that
you don’t think
it’s an “A”!
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Hypotenuse
T an
Opposite
Adjacent
Example: Solve for the length of the hypotenuse
Trigonometry Example
25°
5.2 m
You know an angle and a length opposite the angle
and you want to know the hypotenuse. Opposite &
hypotenuse are used in “sin”
5 2
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hypotenuse
5.2m
sin25 =Hypotenuse *
* Hypotenuse
sin 25°
sin 25°
sin25
5.2m
hypotenuse = hypotenuse=12m

Solving for an Angle
If you know the two sides of the triangle, use the “arc”
trig function to solve for the angle
The “arc” is symbolized with
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= −
hypotenuse
opposite
sinθ 1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= −
hypotenuse
adjacent
cosθ 1
y
“-1” exponent and is found as
a “2nd” function on your
calculator
tetoθ
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⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= −
adjacent
opposite
tanθ 1
Right angle
θ
“Theta θ” - angle in question
Opposit
Adjacent to θ
Example: Solve for the unknown angle
Trigonometry Example - Solving for Angle
?
You know the sides opposite and adjacent to an
unknown angle. “tan-1” can be used
3.7 m
7.5 m
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3.7m
7.5m
tanθ 1−
=
63.7θ =

Definition - Vector
Vectors – Quantities with a magnitude
and a direction.
Trigonometry is used when
calculating with vectors
47/67
calculating with vectors..
Quadratic
Equation
The next topic will be the
d ti ti
48/67
quadratic equation.

Definition – Quadratic Formula
Quadratic Formula – n. formula used
to find “x” in an equation that
contains “x” and “x2” .
4acbb
x
2
−±−
=
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2a
0cbxaxwhere 2
=++
Example
When working velocity problems, you may end up
with equations that contain an “x” and an “x2
Example: Solve for x (“x” represents “time”)
2a
4acbb
x
2
−±−
=
085x1x2
=−+
1*2
8)*1*(455
x
2
−−±−
=
From above: a = 1, b = 5, c = -8
50/67
12
2
575
x
±−
=
Using the + give x = 1.27
Using the – gives x = -6.27
A negative answer isn’t possible for time

Graphing
Graphing is commonly used in
51/67
physics.
Graphing and Slopes
Graphing is used extensively in physics to
visualize and analyze data. Slopes can give
meaningful information.
Y-axis
Position
Velocity
X-axis
Time
Time
Slope
Velocity
Acceleration
52/67
In Calculus, these are called derivatives…Velocity is the
derivative of position.

Finding Average Slopes
Use the final and initial conditions to find the average
slope.
80
0
10
20
30
40
50
60
70
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0
0 2 4 6 8 10
Initial conditions: x = 0 and y = 3
Final conditions: x = 8 and y = 67 12
12
xx
yy
slope
−
−
= 8
08
367
slope =
−
−
=
Average slope = 8
Finding Instantaneous Slopes
Use tangents to find the slope at an instant on a curve
Find slope at x = 5
10
20
30
40
50
60
70
80
54/67
0
10
0 2 4 6 8 10
Initial conditions: x = 5 and y = 28
Final conditions: x = 7.7 and y = 53 12
12
xx
yy
slope
−
−
= 9.3
57.7
2853
slope =
−
−
=
Instantaneous slope at x = 5 is 9.3

35
Graphing Direct Relationships
Many relationships in physics are direct
0
5
10
15
20
25
30
0 2 4 6 8 10 12
Position
Slope is a
straight line
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Time
Equation of line is in format: y = mx + b
where: m = slope
b = y-intercept
Graphing Quadratic Relationships
Some relationships in physics are quadratic
80
0
10
20
30
40
50
60
70
56/67
0
0 2 4 6 8 10
Equation of line is in format: y = kx2 + b
where: k = a constant
b = y-intercept

Graphing Inverse Relationships
Some relationships in physics are inverse
2 9
3
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
57/67
0 1 2 3 4 5 6 7 8
Equation of line is in format: y = k (1/x)
where: k = a constant
Calculator Survival
Finally, we will discuss important
i f i l l
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tips for using your calculator
correctly.

Calculator Survival
Many times a student has a problem set up
correctly, but makes a mistake entering it into the
calculator.
59/67
The following slides will show how to avoid
common calculator mistakes.
Calculator Tip #1
Dividing by more than one number
L)atm)(2.5(1.25
K)L)(273atm)(1.0(1.00
T2 =Example:
Common mistake:
Typing in * tells the calculator that the number
following it is on top of the expression.
1.00 * 1.0 * 273 ÷ 1.25 * 2.5Entering in:
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Correct method:
1.00 * 1.0 * 273 ÷ 1.25 ÷ 2.5Entering in:
Always use ÷ to designate a number on the
bottom.

Calculator Tip #2
Using scientific notation
2
3
10*9.8
10*1.5
D =Example:
Common mistake:
1.5 * 10 ^ 3 ÷ 9.8 * 10 ^ 2Entering in:
Typing in * tells the calculator that the number
following it is on top of the expression.
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Correct method:
1.5 EE 3 ÷ 9.8 EE 2Entering in:
Always use EE (or EXP) key to enter in
Calculator Tip #3
Adding/subtracting & multiplying/dividing together
100*
1.37
1.371.25
error%
−
=Example:
Common mistake:
1.25 - 1.37 ÷ 1.37 * 100Entering in:
Always use parenthesis around addition
Addition and subtraction must happen
first…before multiplication and division.
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Correct method:
(1.25 - 1.37) ÷ 1.37 * 100Entering in:
Always use parenthesis around addition
and subtraction when combining with other
operations.

Calculator Tip #4
Using powers with negative numbers
2
10−Example:
Common mistake:
(-10)2Entering in:
To make something negative keep the
Anything inside the parenthesis will be
squared as well.
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Correct method:
-(102)Entering in:
To make something negative, keep the
negative sign out of the exponent operation.
Or -102
Summary of Calculator Tips
S i tifi t ti Al EE ( EXP) k t t
Dividing by more than one number: Always use ÷ to
designate a number on the bottom
1
Powers & negative numbers: To make something
Combining operations: Always use parenthesis around
addition and subtraction when combining with other
operations
3
Scientific notation: Always use EE (or EXP) key to enter
in scientific notation
2
64/67
Powers & negative numbers: To make something
negative, keep the negative sign out of the exponent
operation
4

Trigonometric
functions are
Trigonometric
functions are
Algebra is
often used to
solve for
Algebra is
often used to
solve for
After working
hard on a
problem, avoid
After working
hard on a
problem, avoid
Learning Summary
used in
physics.
used in
physics.
solve for
variables in
physics.
solve for
variables in
physics.
common
calculator
mistakes!
common
calculator
mistakes!
Results ofResults of
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Follow rules for
working with
exponents &
Follow rules for
working with
exponents &
Results of
calculations must
be reported with
the correct # of sig
figures.
Results of
calculations must
be reported with
the correct # of sig
figures.
Congratulations
You have successfully completed
the core tutorial
Basic Math for Physics
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Basic Math for Physics

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