Mathematics Review For Mathematics Review For PhysicsPhysics

Physics 1 Physics 1

Willowridge H.S.Willowridge H.S.

MATHEMATICSMATHEMATICS is an essential is an essential tool to the tool to the scientist or scientist or engineer. This engineer. This chapter is a chapter is a review of most review of most of the skills that of the skills that are necessary are necessary for for understanding understanding and applying and applying physics. A physics. A thorough thorough review is review is essential.essential.

Preparatory MathematicsPreparatory Mathematics

Note:Note: This module This module maymay be skipped be skipped based on individual needs of the user.based on individual needs of the user.

Basic geometry, algebra, formula Basic geometry, algebra, formula rearrangement, graphing, rearrangement, graphing, trigonometry, scientific notation, and trigonometry, scientific notation, and such are normally assumed for such are normally assumed for beginning physics.beginning physics.

If unsure, please at least run through If unsure, please at least run through the very focused review in this the very focused review in this module.module.

Objectives: After completing Objectives: After completing this module, you should be this module, you should be able to:able to:• AddAdd, , subtractsubtract, , multiplymultiply, and , and dividedivide signed signed

measurements.measurements.

• Solve and evaluate simple Solve and evaluate simple formulasformulas for all for all parameters in an equation.parameters in an equation.

• Work problems in Work problems in Scientific NotationScientific Notation..

• Construct and evaluate Construct and evaluate graphsgraphs..

• Apply rules of Apply rules of geometrygeometry and and trigonometrytrigonometry..

Addition of Signed Addition of Signed NumbersNumbers

• To add two numbers of To add two numbers of like signlike sign, sum , sum the absolute values of the numbers and the absolute values of the numbers and give the sum the common sign.give the sum the common sign.

• To add two numbers of unlike sign, find the difference of their absolute values and give the sign of the larger number.

Example:Example: Add (-6) to (-3) Add (-6) to (-3) (-3) + (-6) = -(3 + 6) = -9 (-3) + (-6) = -(3 + 6) = -9

Example:Example: Add (-6) to (+3). Add (-6) to (+3). (+3) + (-6) = -(6 - 3) = -3(+3) + (-6) = -(6 - 3) = -3

Arithmetic: Come on, Arithmetic: Come on, man . . .man . . .

What’s up with this! I What’s up with this! I have no trouble with have no trouble with addition and addition and subtraction. This is subtraction. This is grade school, man!grade school, man!

Example 1.Example 1. A force directed to the A force directed to the right is positive and a force to the left is right is positive and a force to the left is negative. What is the sum of A + B + C negative. What is the sum of A + B + C if if AA is is 100 lb100 lb, , rightright; ; BB is is 50 lb50 lb, , leftleft; and ; and CC is is 30 lb30 lb, , leftleft..

Given:Given: A = + 100 lb; B = - 50 lb; C = -30 lb

A + B + C = (100 lb) + (-50 lb) + (-30 lb)

A + B + C = (100 lb) + (-50 lb) + (-30 lb)

A + B + C = +20 lbA + B + C = +20 lb

Net Force = 20 lb, rightNet Force = 20 lb, right

100 lb-30 lb

-50 lb

A + B + C = +(100 lb - 50 lb - 30 lb)

Subtraction of Signed Subtraction of Signed NumbersNumbers• To subtract one signed number b from

another signed number a, change the sign of b and add it to a, using the addition rule.

Examples: Subtract (-6) from (-3):

(-3) - (-6) = -3 + 6 = +3 Subtract (+6) from (-3):Subtract (+6) from (-3):

(-3) - (+6) = -3 - 6 = -9(-3) - (+6) = -3 - 6 = -9

Example 2.Example 2. On a winter day, the On a winter day, the temperature drops from temperature drops from 151500CC to a low of to a low of -10-1000CC. What is the change in . What is the change in temperature?temperature?Given:Given: t0 = + 150C; tf = - 100C

150C

-100C

t = tf - t0

t = (-100C) - (+150C) = -100C - 150C = -25 C0

t = -25 C0t = -25 C0

What is the change in temperature if it rises back to +150C? t = +25 C0t = +25 C0

Multiplication: Signed Multiplication: Signed NumbersNumbers

(-12)(-6) = +72 ; (-12)(+6) = -72(-12)(-6) = +72 ; (-12)(+6) = -72

• If two numbers have If two numbers have like signslike signs, their , their product is product is positivepositive..

• If two numbers have If two numbers have unlike signsunlike signs, , their product is their product is negativenegative..

Examples:Examples:

Division Rule for Signed Division Rule for Signed NumbersNumbers

( 72) (-72)12; 12

( 6) (+6)

( 72) (-72)12; 12

( 6) (+6)

• If two numbers have If two numbers have like signslike signs, their , their quotient is quotient is positivepositive..

• If two numbers have If two numbers have unlike signsunlike signs, , their quotient is their quotient is negativenegative..

Examples:Examples:

Extension of Rule for Extension of Rule for FactorsFactors

Examples:Examples:

• The result will be positive if all factors are positive or if there is an even number of negative factors.

• The result will be negative if there is an odd number of negative factors.

( 2)( 4) (-2)(+4)(-3)4 ; 12

2 (-2)(-1)

( 2)( 4) (-2)(+4)(-3)4 ; 12

2 (-2)(-1)

Example 3:Example 3: Consider the following Consider the following formula and evaluate the formula and evaluate the expression for expression for xx when when a = -1a = -1, , b = -b = -22, , c = 3c = 3, , d = -4d = -4..

2cbax cd

bc

2(3)( 2)( 1)(3)( 4)

( 2)(3)x

x = -1 + 48 x = +47x = +47

Working With Formulas:Working With Formulas:Many applications of physics require one to Many applications of physics require one to solve and evaluate mathematical solve and evaluate mathematical expressions called expressions called formulasformulas..

LW

H

Consider Consider Volume VVolume V, for , for example:example:

V = LWHV = LWH

Applying Applying laws of algebralaws of algebra, we can solve for , we can solve for LL, , WW, , or or HH::

VL

WHV

LWH

VW

LHV

WLH

VH

LWV

HLW

Algebra ReviewAlgebra Review

A A formula formula expresses an expresses an equalityequality, and , and that equality must be maintained.that equality must be maintained.

If x + 1 = 5 then x must be equal to 4 in order to maintain equality.

If x + 1 = 5 then x must be equal to 4 in order to maintain equality.

Whatever is Whatever is done to one done to one side of an side of an equation must equation must be done to the be done to the other in order other in order to main-tain to main-tain equality.equality.

For example:• Add or subtract the same

value to both sides.• Multiply or divide both

sides by the same value.• Square or take the square

root of both sides.

For example:• Add or subtract the same

value to both sides.• Multiply or divide both

sides by the same value.• Square or take the square

root of both sides.

Algebra With EquationsAlgebra With EquationsFormulas can be solved by performing a Formulas can be solved by performing a sequence of identical operations to both sides sequence of identical operations to both sides of an equality.of an equality.

• Terms may be added or subtracted from each side of an equality.

x = 2 - 4 + 6

x = +4x = +4

- 4 + 6 = -4 + 6Subtract 4 and Subtract 4 and add 6 to each add 6 to each

sideside

x + 4 - 6 = 2 (Example)

Equations (Cont.)Equations (Cont.)

• Each term on both sides can be multiplied or divided by the same factor.

5x4; 4 5; 20

5 5

xx

5 155 15; ; 3

5 5

xx x

2x 6 42 6 4; ; 3 2; 5

2 2 2x x x

Equations (Cont.)Equations (Cont.)

• The same rules can be applied to literal equations (sometimes called formulas).

2 1

: ( )

FSolved for g g

m m

2 1

: ( )

FSolved for g g

m m

2 1F m g m g Solve for g:

Isolate g by factoring: 2 1( )F g m m Divide both sides by: (m2 – m1)

Equations (Cont.)Equations (Cont.)

• Now look at a more difficult one. (All that is necessary is to isolate the unknown.).

2

; solve for mv

F mg gR

2

2 2 : mv mv

Subract F mgR R

2

2 :

F vDivide by m g

m R

2

2 :

F vSolved for g g

m R

2

2 :

F vSolved for g g

m R

Equations (Cont.)Equations (Cont.)

• Each side may be raised to a power or the root may be taken of each side.

2

; solve for mv

F mg vR

2

2Subract mg:

mvF mg

R

22 2 2Divide by m;multiply by R :

FRgR v

m

22 :

FRSolved for v v gR

m

22 :

FRSolved for v v gR

m

This is Getting Tougher!This is Getting Tougher!Man . . . Arithmetic is Man . . . Arithmetic is one thing, but I gotta one thing, but I gotta have help with solving for have help with solving for those letters.those letters.

Formula RearrangementFormula RearrangementConsider the following Consider the following

formula:formula:A C

B D

Multiply by B to solve for Multiply by B to solve for A:A:

BA BC

B D

Notice that Notice that BB has has moved moved up to the rightup to the right..

1

A BC

D

BCA

DBC

AD

Thus, the solution Thus, the solution for for AA becomes: becomes:

Next Solve for Next Solve for “D”“D”

A C

B D

1. Multiply by “D” 1. Multiply by “D” 2. Divide by “A”2. Divide by “A”

3. Multiply by “B” 3. Multiply by “B” 4. Solution for 4. Solution for “D” “D”

DD moves up to moves up to left.left.

AA moves down to moves down to right.right.B B moves up to moves up to

right.right.DD is then isolated. is then isolated.

DA DC

B D

DA C

AB A

BD BC

B A

BCD

A

Cross Roads for FactorsCross Roads for FactorsWhen there are only When there are only two termstwo terms in a formula in a formula separated by an equals sign, the separated by an equals sign, the cross roadscross roads can can be used.be used.

AB DE

C F

Cross RoadsCross Roads for Factors for Factors

Only!Only!

Example solutions are given below:Example solutions are given below:

1

A CDE

BF

1

F CDE

AB

1

ABF D

CE

Example 4:Example 4: Solve for Solve for n.n.

PV = nRT PV nRT

1 1=

R T=

PV n

1R T

R T

PVn

RTPV

nRT

=PV n

1R T

CAUTION SIGNS CAUTION SIGNS FOR CROSS ROADSFOR CROSS ROADS

The The cross-roadcross-road method method works ONLY for works ONLY for

FACTORS!FACTORS!( )a b c e

d f

The “c” cannot be moved unless the entire factor (b + c) is

moved.

Solution for Solution for aa:: ( )

eda

b c f

( )

eda

b c f

CautionCaution

Example 5:Example 5: Solve for Solve for f.f.( )a b c e

d f

First move First move ff to get it in numerator. to get it in numerator.

( )a b c ef

d

Next moveNext move aa, d, and (b + c), d, and (b + c)

( )

edf

a b c

( )

edf

a b c

When to use Cross-Roads:When to use Cross-Roads:

1. Cross roads works only when 1. Cross roads works only when a formula has ONE term on a formula has ONE term on each side of an equality.each side of an equality.

2. Only FACTORS may be 2. Only FACTORS may be moved!moved!

AB DE

C F

WARNING: DON’T SHOW WARNING: DON’T SHOW THIS “CROSS ROADS” THIS “CROSS ROADS” APPROACH TO A MATH APPROACH TO A MATH

TEACHER!TEACHER!

Use the technique because it works and is effective.

Recognize the problems of confusing factors with terms.

Use the technique because it works and is effective.

Recognize the problems of confusing factors with terms.

BUT . . . Don’t expect all instructors to like it. Just use it quietly, and don’t tell anyone.

BUT . . . Don’t expect all instructors to like it. Just use it quietly, and don’t tell anyone.

Often It is Necessary to Use Often It is Necessary to Use Exponents in Physics Exponents in Physics

Applications.Applications.E = E = mcmc22

E = m ( c E = m ( c c )c )

The exponent “2” The exponent “2” means “ c” times means “ c” times

“c”“c”

E = E = mcmc22

Cube of side x

Volume of a cube of Volume of a cube of side x is “x x x” orside x is “x x x” or

V = xV = x33

Bumpy Road Ahead !Bumpy Road Ahead !

Rules for Exponents Rules for Exponents and Radicals are and Radicals are difficult to apply—but difficult to apply—but necessary in physics necessary in physics notation.notation.Please fight your Please fight your way through this way through this review—ask for review—ask for help if needed.help if needed.

Exponents and Radicals Exponents and Radicals Multiplication RuleMultiplication Rule

When two quantities of the same base When two quantities of the same base are multiplied, their product is obtained are multiplied, their product is obtained by adding the exponents algebraically.by adding the exponents algebraically.

When two quantities of the same base When two quantities of the same base are multiplied, their product is obtained are multiplied, their product is obtained by adding the exponents algebraically.by adding the exponents algebraically.

( )( )m n m na a a ( )( )m n m na a a

ExampleExamples:s:

3 5 3 5 82 2 2 2 4 1 5 6x x x x

Exponent RulesExponent Rules

Negative Exponent:Negative Exponent: A term that is not A term that is not equal to zero may have a negative equal to zero may have a negative exponent as defined below:exponent as defined below:

Negative Exponent:Negative Exponent: A term that is not A term that is not equal to zero may have a negative equal to zero may have a negative exponent as defined below:exponent as defined below:

1 1 or n n

n na a

a a

1 1

or n nn n

a aa a

ExampleExamples:s:

22

12 0.25

2

2 3 3 4 7

4 2 2

x y y y y

y x x

Exponents and Exponents and Radicals Zero Radicals Zero

ExponentExponentZero Exponent:Zero Exponent: Any quantity raised Any quantity raised to the power of zero is equal to 1.to the power of zero is equal to 1.Zero Exponent:Zero Exponent: Any quantity raised Any quantity raised to the power of zero is equal to 1.to the power of zero is equal to 1.

0The zero exponent: 1a 0The zero exponent: 1a

YES, that’s rightYES, that’s right

ANYTHING ! ANYTHING ! Raised to the Raised to the zero power is zero power is

“1”“1”

01

Exponents and Exponents and Radicals Zero Radicals Zero

ExponentExponentZero Exponent:Zero Exponent: Consider the Consider the following examples for zero following examples for zero exponents.exponents.

Zero Exponent:Zero Exponent: Consider the Consider the following examples for zero following examples for zero exponents.exponents.

0The zero exponent: 1a 0The zero exponent: 1a

0 3 0 3x y z y

03

4

0

10.333

3 3

xy

z

Other Exponent RulesOther Exponent Rules

Division rule:Division rule: When two When two quantities of the same base quantities of the same base are divided, their quotient is are divided, their quotient is obtained by subtracting the obtained by subtracting the exponents algebraically.exponents algebraically.

Division rule:Division rule: When two When two quantities of the same base quantities of the same base are divided, their quotient is are divided, their quotient is obtained by subtracting the obtained by subtracting the exponents algebraically.exponents algebraically.

mm n

n

aa

a

Example:Example:4 2 4 1 2 5 3 3 3

5 31 1

x y x y x y x

xy y

Exponent Rules Exponent Rules (Continued):(Continued):

Power of a Power:Power of a Power: When a When a quantity quantity aamm is raised to the is raised to the power power mm::

Power of a Power:Power of a Power: When a When a quantity quantity aamm is raised to the is raised to the power power mm::

nm mna a

Examples:Examples:

3 5 (5)(3) 15 2 3 6( ) ; ( )x x x q q

Exponent Rules Exponent Rules (Continued):(Continued):

Power of a product: Obtained by applying the exponent to each of the factors.

Power of a product: Obtained by applying the exponent to each of the factors.

Example:Example:12

3 2 4 (3)(4) ( 2)(4) 12 88

( )x

x y x y x yy

m m mab a b

Exponent Rules Exponent Rules (Continued):(Continued):

Power of a quotient:Power of a quotient: Obtained Obtained by applying the exponent to by applying the exponent to each of the factors. each of the factors.

Power of a quotient:Power of a quotient: Obtained Obtained by applying the exponent to by applying the exponent to each of the factors. each of the factors.

n n

n

a a

b b

Example:Example:33 2 9 6 9 9

3 3 9 3 6

x y x y x p

qp q p q y

Roots and RadicalsRoots and Radicals

Roots of a product:Roots of a product: The The nnth th root of a product is equal to root of a product is equal to the product of the the product of the nnth roots th roots of each factor.of each factor.

Roots of a product:Roots of a product: The The nnth th root of a product is equal to root of a product is equal to the product of the the product of the nnth roots th roots of each factor.of each factor.

n n nab a b

3 3 38 27 8 27 2 3 6

Example:Example:

Roots and Radicals (Cont.)Roots and Radicals (Cont.)

Roots of a Power:Roots of a Power: The The roots of a power are found roots of a power are found by using the definition of by using the definition of fractional exponents:fractional exponents:

Roots of a Power:Roots of a Power: The The roots of a power are found roots of a power are found by using the definition of by using the definition of fractional exponents:fractional exponents:

/n m m na a

16 12 16/ 4 12/ 4 4 34 x y x y x y

Examples:Examples:

6 3 6/3 3/3 2

39 9/3 3

x y x y x y

z z z

Scientific NotationScientific Notation

0 000000001 10

0 000001 10

0 001 10

1 10

1000 10

1 000 000 10

1 000 000 000 10

9

6

3

0

3

6

9

.

.

.

, ,

, , ,

Scientific notationScientific notation provides a short-hand method for expressing provides a short-hand method for expressing very small and very large numbers.very small and very large numbers.

Examples:

93,000,000 mi = 9.30 x 107 mi

0.00457 m = 4.57 x 10-3 m

2

-3

876 m 8.76 x 10 m

0.0037 s 3.7 x 10 sv

53.24 x 10 m/sv 53.24 x 10 m/sv

GraphsGraphs

Direct relationshipDirect relationship

Increasing values on the horizontal axis cause a proportional increase in values on the vertical axis.

Increasing values on the horizontal axis cause a proportional increase in values on the vertical axis.

Increasing values on the horizontal axis cause a proportional decrease in values on the horizontal axis.

Increasing values on the horizontal axis cause a proportional decrease in values on the horizontal axis.

Indirect relationshipIndirect relationship

GeometryGeometryAnglesAngles are measured in terms are measured in terms of degrees, from of degrees, from 0°0° to to 360360ºº..

Line Line ABAB is is perpendicularperpendicular to line to line CDCD

A

B

C D

AB CDAB CD

270º

180º 0º, 360º

90º

Angle

A

B

C

D

Line Line ABAB is is parallelparallel to to line line CDCD

AB CDAB CD

Geometry (Continued)Geometry (Continued)When When two straight two straight lines intersectlines intersect, they , they form opposing angles form opposing angles which are equal.which are equal.

A A B

B

Angle A = Angle A

Angle B = Angle B

Angle A = Angle A

Angle B = Angle B

When a When a straight line straight line intersects two parallel intersects two parallel lineslines, the alternate interior , the alternate interior angles are equal.angles are equal.

A

A

B B

Angle A = Angle A

Angle B = Angle B

Angle A = Angle A

Angle B = Angle B

Geometry (Cont.)Geometry (Cont.)For any triangle, the For any triangle, the sum of the interior sum of the interior angles is 180ºangles is 180º

For any right For any right triangle, the triangle, the sum sum of the two smaller of the two smaller angles is 90ºangles is 90º

A + B + C = 180°A + B + C = 180°

AC

B

A + B = 90°A + B = 90°

AC

B

Example 6:Example 6: Use geometry to Use geometry to determine the unknown angles determine the unknown angles and and in figure.in figure.1. Draw helping lines 1. Draw helping lines

ABAB and and CDCD..

500200

2. Note:2. Note: + 50 + 5000 = = 909000

A BC

D

3. Alternate, interior angles are equal:

4. ACD is a right angle:

200 + + 400 = 900

Right Triangle TrigonometryRight Triangle TrigonometryAngles are often represented by Greek letters:

alpha beta gamma

theta phi delta

Angles are often represented by Greek letters:

alpha beta gamma

theta phi delta

Pythagorean theoremPythagorean theorem

The square of the The square of the hypotenuse is equal to hypotenuse is equal to the sum of the squares the sum of the squares of the other two sides.of the other two sides.

Pythagorean theoremPythagorean theorem

The square of the The square of the hypotenuse is equal to hypotenuse is equal to the sum of the squares the sum of the squares of the other two sides.of the other two sides.

R

x

y

2 2 2

2 2

R x y

R x y

2 2 2

2 2

R x y

R x y

Right Triangle Right Triangle TrigonometryTrigonometry

hyp

adj

opp

The The sinesine value of a right triangle value of a right triangle is equal to the ratio of the length is equal to the ratio of the length of the side of the side opposite opposite the angle to the angle to the length of the the length of the hypotenusehypotenuse of the of the triangle.triangle.

The The cosine cosine value of a right triangle is equal to the value of a right triangle is equal to the ratio of the length of the side ratio of the length of the side adjacentadjacent to the angle to the angle to the length of the to the length of the hypotenusehypotenuse of the triangle. of the triangle.

The The tangenttangent value of a right triangle is equal to value of a right triangle is equal to the ratio of the length of the side the ratio of the length of the side oppositeopposite the the angle to the side angle to the side adjacentadjacent to the angle. to the angle.

sinOpp

Hyp

cosAdj

Hyp

tanOpp

Adj

Example 5:Example 5: What is the distance What is the distance xx across the lake, and what is the across the lake, and what is the angle angle ??

x

20 m

12 m

R = 20 m is hypotenuse. R = 20 m is hypotenuse. Thus, Pythagoras’ Theorem:Thus, Pythagoras’ Theorem:

2 2 2(20) (12)x

400 144 256x

x = 16 mx = 16 m

12 mcos

20 m

adj

hyp 53.1

0

53.10

SummarySummary

• To add two numbers of To add two numbers of like signlike sign,, sum sum the absolute values of the numbers the absolute values of the numbers and give the sum the common sign.and give the sum the common sign.

• To add two numbers of unlike sign, find the difference of their absolute values and give the sign of the larger number.

• To To subtractsubtract one signed number one signed number bb from from another signed number another signed number aa, change the , change the sign of sign of bb and add it to and add it to aa, using the , using the addition rule.addition rule.

Summary (Cont.)Summary (Cont.)

• If two numbers have If two numbers have unlike signsunlike signs, , their product is their product is negativenegative..

• If two numbers have If two numbers have like signslike signs, , their product is their product is positivepositive..

• The result will be positive if all factors are positive or if there is an even number of negative factors.

• The result will be negative if there is an odd number of negative factors.

SummarySummary

1nn

aa

1nn

aa

0 1a 0 1a

1nn

aa1nn

aa

mm n

n

aa

a

mm n

n

aa

a nm mna a nm mna a( )( )m n m na a a ( )( )m n m na a a

Working with Equations:

• Add or subtract the same value to both sides.

• Multiply or divide both sides by the same value

• Square or take the square root of both sides.

Working with Equations:

• Add or subtract the same value to both sides.

• Multiply or divide both sides by the same value

• Square or take the square root of both sides.

Summary (Cont.)Summary (Cont.)

m m mab a bn n

n

a a

b b

n n nab a b

/n m m na a

Review sections on Review sections on scientific notationscientific notation, , geometrygeometry, , graphsgraphs, and , and trigonometrytrigonometry as needed. as needed.

Review sections on Review sections on scientific notationscientific notation, , geometrygeometry, , graphsgraphs, and , and trigonometrytrigonometry as needed. as needed.

Trigonometry ReviewTrigonometry Review

• You are expected to know the You are expected to know the following: following:

y

x

R

y = R sin y = R sin

x = R cos x = R cos

siny

R

cosx

R

tany

x R2 = x2 +

y2

R2 = x2 + y2

TrigonometryTrigonometry

Conclusion of MathematicsConclusion of Mathematics