basic math for physics

AP Physics Rapid Learning Series - 02

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Basic Math for as c at oPhysics

Physics Rapid Learning Series

Rapid Learning Centerwww.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 ConclusionsResults &

Conclusions

Data taken with correct # of sig figs

Data taken with correct # of sig figs

CalculationsCalculationsFollow 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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Quadratic Quadratic FormulaTrigonometryTrigonometryAlgebra

Rules for

exponents

Rules for working with

exponents

Use scientific Use 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:

Exampleunknown variable

sides

Added Subtract

Example

5 = x + 2-2 -25-2 = x

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Subtracted Add3 = x – 6

+6 +63+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 = 4x4 4

2÷4 = x

unknown variable

sides

Multiplied Divide

Example

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Divided Multiply2* 6 = x *2

22*6 = x

Example: VelocityVelocity problems require algebraic work:

displacement(d)Velocitytime(t)

=

Example: Solve for distance

The unknown variable (d) is being divided by 25 s

time(t)

25sdm/s52 = * 25 s25 s *

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The unknown variable (d) is being divided by 25 s.

Multiply both sides by 25 sdm/s52*25s =

d1300m =



Example: ForceNet 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 ForceWhen 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( ) 2d

/kgNm106.6N101.7 ×=×

( ) 222118

dg)(50kg)(50k/kgNm106.6N101.7 −− ×=×d2 * * d2

( ) ( ) g)(50kg)(50k/kgNm106.6N101.7d 221182 −− ×=×Solve for d

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( ) ( )1.7x10-8 N 1.7x10-8 N

N)10(1.7kg))(50kg)(50/kgNm10(6.6d 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 & SubtractingAdding & subtracting:

Perform the calculation1

Example: Perform the following operation:10.027 g

1 5 g3 decimal places1 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 g8.527 g

1 decimal place

8.5 g

1

Least = 1

2

3



Rounding: Multiplying & DividingMultiplying & Dividing:

Perform the calculation1

Example: Perform the following operation:

Determine the least # of significant figures in the problemRound the answer to that many significant figures.

2

3

110.027 m

6 68467 m/s5 sig figures

=

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6.68 m/sLeast = 3

23

1.50 s6.68467 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 NotationRules 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 g1


“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 1.027500456The original number was “big” so it will be a +6 power

1.027500456 x 106 g

12

3

123456



Writing in Scientific Notation 2Rules for writing scientific notation:

Decimal is moved to follow the 1st non-zero number


1

2

Examples: Write in scientific notation:

0.0007543 g1


“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 0007.543The original number was “small” so it will be a -4 power

7.543 x 10-4 g

12

3

1234

Sig Figures & Scientific NotationScientific Notation can also be used to write a specific number of significant figures.

Example: Your calculator gives you: 10200 gAnd you need to write the result with 4 significant figures

10200 g 3 significant figures

10200. g 5 significant figures4321

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1.020 X 104 g 4 significant figures!

Use scientific notation

4321



Reading Scientific Notation - 1Rules 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) numbers2

3.25 X 10-6 g00000123456

.

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12

The decimal point will move 6 timesThe power is negative, so we’ll make the number “small”—move the decimal forward (left)

0.00000325 g

123456

Reading Scientific Notation - 2Rules 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) powers2

7.2004 X 103 mL.123

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12

The decimal point will move 3 timesThe 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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gcalculations 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 yare 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 yare different!

Example:158

123 (1512)

(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 =192

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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 exponents 5

Dividing: subtract the exponents. 32÷38 = 3-64

Multiplying: add the exponents. 32*38 = 3103

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 = 365

Calculations with Scientific Notation

There are several e e a e se e asimilar 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 pof 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 pof 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

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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

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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)

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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 FunctionsThe three common trig functions work with right triangles.

itTrig functions use one of the

hypotenuseoppositesinθ =

hypotenuseadjacentcosθ =

Trig functions use one of the non-90° angles and discuss the sides of the triangle as they relate to that angle

te to

θ

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adjacentoppositetanθ =

Right angle θ“Theta”—angle in question

Opp

osit

Adjacent to θ



Remembering the Functions

S inO pposite

Make sure youH ypotenuseC osAdjacentHypotenuse

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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HypotenuseT anOppositeAdjacent

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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hypotenuse5.2msin25 =Hypotenuse * * Hypotenuse

sin 25°sin 25°

sin255.2mhypotenuse = hypotenuse=12m



Solving for an AngleIf you know the two sides of the triangle, use the “arc” trig function to solve for the angle

The “arc” is symbolized with

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

hypotenuseoppositesinθ 1

⎟⎟⎠

⎞⎜⎜⎝

⎛= −

hypotenuseadjacentcosθ 1

y“-1” exponent and is found as a “2nd” function on your calculator

te to

θ

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⎟⎟⎠

⎞⎜⎜⎝

⎛= −

adjacentoppositetanθ 1

Right angleθ“Theta θ” - angle in question

Opp

osit

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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3.7m7.5mtanθ 1−=

63.7θ =



Definition - Vector

Vectors – Quantities with a magnitude and a direction.

Trigonometry is used when calculating with vectors

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calculating with vectors..

Quadratic Equation

The next topic will be the d ti ti

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quadratic equation.



Definition – Quadratic Formula

Quadratic Formula – n. formula used to find “x” in an equation that contains “x” and “x2” .

4acbbx2 −±−

=

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2a

0cbxaxwhere 2 =++

ExampleWhen working velocity problems, you may end up with equations that contain an “x” and an “x2

Example: Solve for x (“x” represents “time”)

2a4acbbx

2 −±−=

085x1x2 =−+

1*28)*1*(455x

2 −−±−=

From above: a = 1, b = 5, c = -8

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12

2575x ±−

=

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

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physics.

Graphing and SlopesGraphing 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

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In Calculus, these are called derivatives…Velocity is the derivative of position.



Finding Average SlopesUse the final and initial conditions to find the average slope.

80

0

10

20

30

40

50

60

70

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00 2 4 6 8 10

Initial conditions: x = 0 and y = 3Final conditions: x = 8 and y = 67 12

12

xxyyslope

−−

= 808367slope =

−−

=

Average slope = 8

Finding Instantaneous SlopesUse tangents to find the slope at an instant on a curve

Find slope at x = 5

10

20

30

40

50

60

70

80

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0

10

0 2 4 6 8 10

Initial conditions: x = 5 and y = 28Final conditions: x = 7.7 and y = 53 12

12

xxyyslope

−−

= 9.357.7

2853slope =−−

=

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

Posi

tion

Slope is a straight line

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Time

Equation of line is in format: y = mx + bwhere: m = slope

b = y-intercept

Graphing Quadratic Relationships

Some relationships in physics are quadratic

80

0

10

20

30

40

50

60

70

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00 2 4 6 8 10

Equation of line is in format: y = kx2 + bwhere: k = a constant

b = y-intercept



Graphing Inverse Relationships

Some relationships in physics are inverse

2 93

22.12.22.32.42.52.62.72.82.9

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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.

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The following slides will show how to avoid common calculator mistakes.

Calculator Tip #1Dividing by more than one number

L)atm)(2.5(1.25K)L)(273atm)(1.0(1.00T2 =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 #2Using scientific notation

2

3

10*9.810*1.5D =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 scientific notation.

Calculator Tip #3Adding/subtracting & multiplying/dividing together

100*1.37

1.371.25error% −=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 #4Using powers with negative numbers

210−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

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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 &

scientific notation.

Follow rules for working with exponents &

scientific notation.

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

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