Math Review Physics 1 DEHS 2011-12 1
Dec 15, 2015
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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:
€
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
2
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