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Math Review

Physics 1DEHS 2011-12

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Math and Physics

• Physics strives to show the relationship between two quantities (numbers) using equations

• Equations show the mathematical relationship between an independent variable and a dependent variable.

• Everything else is regarded as a constant

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Variables• Dependent Variable: is the observed

phenomenon• Independent variable: is the controlled or

selected by the experimenter to determine the relationship to the dependent variable

• Example: You are analyzing the motion of a car and you want to investigate how the car’s distance from start varies with time. Time is the independent variable and distance is the dependent variable

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Variable Notion

• You could pick any symbol to represent any quantity you wish, but there are widely used ways to represent certain quantities

• Most of the time they make sense (m stands for mass, F stands for force), but sometimes we just use an arbitrarily selected, traditional letter (p stands for momentum, J stands for impulse)

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Variable Notion• Sometimes we use letters from the Greek

alphabet. Commonly used are:– Δ = “Delta”, Σ = “Sigma”, θ = “Theta”, μ = “Mu”

• Sometimes the same quantity is used in special circumstances, here we use a subscript to distinguish– Written smaller and lower– Example: vf is final velocity and vi is initial velocity;

FN is normal force and Ff is friction force

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Δ = “change in” or “difference between”

• When you see a Δ in front of a variable, it means “change in” or “difference between” the value of that quantity at two different times/places

• To calculate Δx, you always take it to mean Final value – Initial value

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Δx = x f − x i

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Algebra: Linear equations• Linear equations are polynomials of order 1– Exponent on the dependent variable is 1

• General form looks like: – y represents the dependent variable– x represents the independent variable– m is the constant number that multiplies x, it is

called the slope– b is called the y-intercept, it shows the value of y

when x = 0

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y = mx + b

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Algebra: Linear equations

• The graph of a linear equation looks like a line– If m > 0 the line will go up (/)– If m < 0 the line will go down (\)– If m = 0 the line will be flat (−)

• To solve follow reverse order of operations– Addition/subtraction– Multiplication/division

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Solving Linear Equations Example 1Solve the following for the independent variable:

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v f = v i − gtIdentify the parts:

vf t -g vi

Put into standard form:

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y = mx + b€

v f = −gt + v i

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Solving for an unknown in the denominator

• To solve for an unknown in the denominator of a term:– Cross multiply– Follow the steps previously discussed

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Unknown in the denominator Ex. 1Solve the following equation for T1:

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V1

T1

=V2

T2

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Solving for an unknown in the denominator: Handy Trick

• If you are solving for the denominator of a fraction that is equivalent to a fraction with a denominator of 1, just trade as shown.– This situation comes up ALOT! This trick with save

you some time.

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b =a

x

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Cancelling Variables• Situations frequently come up where one

variable can be dropped from the equation– Recognizing these situations can save you some

work• A variable can only be cancelled when it is in

every term

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12mv i

2 + mghi = 12mv f

2 + mgh f

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Solving Quadratic Equations

• A quadratic equation is a second degree polynomial equation

• It is of the form (or can be manipulated to look like: Ax2 + Bx + C = 0

• There are three common ways of solving– If B = 0 it is easiest to use the _________________– If B ≠ 0, you can use graphical techniques or use the

___________________

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Solving w/ Sq. Rt. Method ExampleSolve the following for f:

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Fc = 4π 2mrf 2

Solve the following for vi:

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v f2 = v i

2 − 2gΔy

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Solving when the unknown is in >1 term

• If the unknown you are solving for is in more than one term (all of the same order) follow these steps:– Add/subtract to get all terms containing your

unknown to the same side– Add/subtract to get all terms not containing your

unknown to the other side– Factor out your unknown– Divide by the quantity multiplying your unknown

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Unknown in >1 term Ex 1

Solve the following for F:

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12 F = μ mg− 1

3 F( )F on right side is inside parenthesis, distribute μ

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Solving when the unknown is in >1 term

• If the unknown you are solving for is in more than two terms and are order 2 and order 1 follow the steps for solving a quadratic eqn:– Put the equation into the general form that looks

like:– Identify A, B, & C– Use the quadratic formula or QUADFORM

program to solve for the unknown– You will usually get two answers, pick the right

one€

Ax 2 + Bx +C = 0

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Unknown in >1 term Ex 2Solve the following for t when Δx = 20, vi = 5 and a = 2

using for the following equation:

Put equation into general form

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Δx = v it +12 at

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Identify your A, B, & CFill in your givens

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Ax 2 + Bx +C = 0

Solve using the quadratic formula or QUADFORM

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Factor Label

• The factor label method (you might remember it from stoichiometry) is used to convert measurements to different units

• Your equation sheet has unit equivalencies• To eliminate a unit on top, put that unit on the

bottom of your factor fraction • To eliminate a unit on bottom, put that unit on

the top of your factor fraction

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Factor Label Ex 1 & 2• Convert 122 cm to m

• Convert 2.3 kg to mg

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Factor Label Ex 3 & 4• Convert 24 m/s to m/min

• Convert 36 km/h to m/s

This is a very common conversion. It may be worth committing the following shortcut to memory: to convert from km/h to m/s divide by 3.6.

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Factor Label Ex 5• It is also worth noting that when converting units

that are raised to some power, require an extra step– 1 m is 100 cm but 1 m2 is NOT 100 cm2

• Convert 0.25 m3 to cm3

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Proportionality – Describing Math

• In physics, we describe the relationship between two quantities as “proportional to __”

• Two quantities are said to be proportional if their ratio is constant

• So A and B are proportional if A=kB or k = A/B– k is called the “constant of proportionality”– if this is true,

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A ~ B (or alternately A∝ B)

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Directly Proportional

• Direct proportionality: The increase in the dependent variable is proportional to the increase in the independent variable

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Δy = k Δx( )

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Proportional to a Power • Direct proportionality (to a power of x):

relationship is described by an equation in which the independent variable is raised to a positive power other than 1– y is proportional to the square of x ( y ~ x2)

– y is proportional to the cube of x ( y ~ x3)

– y is proportional to the square root of x ( y ~ x1/2)

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Inversely Proportional

• Inverse proportionality: The increase in the dependent variable is proportional to the decrease in the independent variable

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Δy =k

Δx( )n

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y ~1

x n

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Graphing

• Graphs help to understand the relationship between two variables

• You will be expected to be able to determine a graph’s general shape just by looking at the equation

The 4 Basic Graph Shapes

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y ~ x

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y ~ x 2 + x

or

y ~ x 2

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y ~1

x n

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y ~ x

or

y 2 ~ x

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Directly Proportional Relationships

• The relationship between two variables is described as being directly proportional if the equation relating the two is linear

– Linear equations have the form:

– The graph of a linear equation is called linear

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y = mx + b

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Parts of a Linear Equation

• m is known as the slope

• Slope is calculated as:

• b is known as the y-intercept • It is calculated by plugging in x = 0 and solving

for y €

m ="rise"

"run"

m =Δy

Δx

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How slope affects the graph

• If m > 0, then the graph will have a slope up

• The greater the value of |m|, the steeper the graph will appear

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Graphing Linear Functions Ex 1

Sketch the graph of

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y = 12 x

y = x

y = 2x

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Graphing Linear Functions Ex 2

Sketch the graph of

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y = x

y = −x

y = −2x

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Graphing Linear Functions Ex 3

Sketch the graph of

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y = x

y = x +1

y = x −1

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Parts of a Quadratic Equation

• Quadratic equations take the form

• A is the coefficient that describes the long-term behavior or y, pay attention to the sign of this term to decide what direction the function goes for large values of x

• B is the coefficient that describes the short-term behavior or y, pay attention to the sign of this term to decide what direction the function goes for small values of x

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y = Ax 2 + Bx +C

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Graphing Quadratic Functions Ex

Sketch the graph of

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y = x 2 + x

y = −x 2 + x

y = x 2 − x

y = −x 2 − x

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Deciding the Graph

• Ignore all other variables in the equation except your independent and dependent variables keep the signs of the variables

• Then match the function to the form of the four basic types of equations

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Deciding the Graph Ex 1

Sketch X vs T graph of the equation:

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x = vt

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Deciding the Graph Ex 2

Sketch Y vs T graph of the equation:

(assume vi > 0 and g > 0)

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Δy = v it −12 gt

2

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Deciding the Graph Ex 3

Sketch V vs X graph of the equation:

(assume vi = 0 and a > 0)

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v f2 = v i

2 + 2aΔx

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Deciding the Graph Ex 4

Sketch F vs m1m2 graph of

the equation:

(assume all numbers are positive and m1 = m2)

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F =Gm1m2

r2

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Deciding the Graph Ex 5

Sketch F vs r graph of the equation:

(assume all numbers are positive)

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F =Gm1m2

r2