Math Review Physics 1 DEHS

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 1

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 2

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) 3

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: v f is final velocity and v i is initial velocity; F N is normal force and F f is friction force 4

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 5

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 6

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 7

Solving Linear Equations Example 1 Solve the following for the independent variable: 8 Identify the parts: v f  t  -g  v i  Put into standard form:

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 9

Unknown in the denominator Ex. 1 Solve the following equation for T 1 : 10

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. 11

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 12

Solving Quadratic Equations A quadratic equation is a second degree polynomial equation It is of the form (or can be manipulated to look like: Ax 2 + 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 ___________________ 13

Solving w/ Sq. Rt. Method Example Solve the following for f: 14 Solve the following for v i :

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 15

Unknown in >1 term Ex 1 Solve the following for F: 16 F on right side is inside parenthesis, distribute μ

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 17

Unknown in >1 term Ex 2 Solve the following for t when Δx = 20, v i = 5 and a = 2 using for the following equation: 18 Put equation into general form Identify your A, B, & C Fill in your givens 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 20

Factor Label Ex 1 & 2 Convert 122 cm to m Convert 2.3 kg to mg 21

Factor Label Ex 3 & 4 Convert 24 m/s to m/min Convert 36 km/h to m/s 22 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.

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 m 2 is NOT 100 cm 2 Convert 0.25 m 3 to cm 3 23

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, 24

Directly Proportional 25 Direct proportionality: The increase in the dependent variable is proportional to the increase in the independent variable

Proportional to a Power 26 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 ~ x 2 ) – y is proportional to the cube of x ( y ~ x 3 ) – y is proportional to the square root of x ( y ~ x 1/2 )

Inversely Proportional 27 Inverse proportionality: The increase in the dependent variable is proportional to the decrease in the independent variable

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 28

The 4 Basic Graph Shapes 29

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 30

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 31

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 32

Graphing Linear Functions Ex 1 Sketch the graph of 33

Graphing Linear Functions Ex 2 Sketch the graph of 34

Graphing Linear Functions Ex 3 Sketch the graph of 35

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 36

Graphing Quadratic Functions Ex Sketch the graph of 37

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 38

Deciding the Graph Ex 1 Sketch X vs T graph of the equation: 39

Deciding the Graph Ex 2 Sketch Y vs T graph of the equation: (assume v i > 0 and g > 0) 40

Deciding the Graph Ex 3 Sketch V vs X graph of the equation: (assume v i = 0 and a > 0) 41

Deciding the Graph Ex 4 Sketch F vs m 1 m 2 graph of the equation: (assume all numbers are positive and m 1 = m 2 ) 42

Deciding the Graph Ex 5 Sketch F vs r graph of the equation: (assume all numbers are positive) 43