Math and Measurements in Physics (or what you should know before beginning AP Physics C)

Scientific Notation A shorthand way to write very large or very small numbers (ex: speed of light, mass of an atom) SI system is based on the powers of ten Expresses numbers multiplied with powers of the number 10: 5,943,000,000 = 5.943 x 109 0.000587 = 5.87 x 10-4 10x 10-1 = 0.1 100 = 1. 101 = 10. 102 = 100. 103 = 1000. 104 = 10000. 1

Algebra Mutiplying Dividing Adding A B C D AC BD A/B C/D AD BC BC In physics, it is usually necessary to solve for unknown quantities that have been written as variables using symbols or letters. Algebraic operations are then used to solve for the unknowns. When working with fractions, here are some rules to remember and follow Mutiplying Dividing Adding A B C D AC BD A/B C/D AD BC x = ± BC 2

Examples – Factor the following 3 1

Differences of Squares Algebra Factoring is an important process in algebra which is used to simplify expressions, simplify fractions, and solve equations. Common Factor ax + ay + az = a (x + y + z) Perfect Square a2 + 2ab + b2 = (a + b)2 Differences of Squares a2 - b2 = (a + b) (a – b) 4

Algebra ax2 + bx + c = 0 The general form of a quadratic equation is: The two roots of this equation are found by using the quadratic formula. 5

y = mx + b Algebra m = Δ y /Δ x = slope of line b = y-intercept A linear equation is an equation for a straight line. The relationship between two variables can often be expressed in terms of a linear equation. y = mx + b m = Δ y /Δ x = slope of line b = y-intercept y Δ y (0,b) Δ x x 6

Examples – Solve the following 7

Algebra Method 1: Substitution 5x + y = -8 2x – 2y = 4 y = -8 +5x Solving Systems of Equations Sometimes there is more than one unknown in an equation. Method 1: Substitution Rearrange 5x + y = -8 2x – 2y = 4 y = -8 +5x Substitute 2x – 2(-8+5x) = 4 Solve x = -1 ; y = -3 8

Algebra 5x + y = -8 5x + y = -8 2x – 2y = 4 x – y = 2 6x = -6 Method 2: Elimination 5x + y = -8 2x – 2y = 4 5x + y = -8 Restate x – y = 2 Add to eliminate 6x = -6 Solve x = -1 ; y = -3 9

Examples Solve the two simultaneous equations below: 10

√ s = rθ Geometry Radian Measure Distance Formula Here are some formulas to review: Distance Formula To find the distance between points (x1, y1) and (x2, y2). d = (x2-x1) + (y2-y1) √ Radian Measure s θ With r the radius of the circle, θ the angle in radians, and s the arc length s = rθ r 11

Geometry Important Formulas Area of a rectangle = length x width Area of a circle = π radius Area of a triangle = ½ base x height Volume of sphere = 4/3 π radius Volume of Rectangular Box = length x width x height Volume of Cylinder = π radius x length Surface Area of sphere = 4 π radius Circumference of a circle = 2π radius 12

Trigonometry Many concepts in physics are analyzed using right triangles. 13

Trigonometry Identities sin θ + cos θ = 1 sec θ = 1 + tan θ sin 2 θ = 2 sin θ cos θ cos 2 θ = cos θ – sin θ tan 2 θ = 2 tan θ/ (1 – tan θ) csc θ = 1 + cot θ sin θ/2 = ½( 1 – cos θ) cos θ/2 = ½( 1+ cos θ) 1 – cos θ = 2 sin θ/2 tan θ/2 = (1-cos θ)/(1+ cos θ) sin (A±B) = sinAcosB ± cosAsinB cos (A±B) = -(cosAcosB± sinAsinB) sin A ± sin B = 2 sin (1/2(A±B))cos(-1/2(A±B)) cos A + cos B = 2 cos (1/2(A+B)cos(1/2(A-B) cos A – cos B = 2 sin (1/2(A+B)sin(1/2(B-A)) 14

Example A person attempts to measure the height of a building by walking out a distance of 46.0 m from its base and shining a flashlight beam toward its top. He finds that when the beam is elevated at an angle of 39 degrees with respect to the horizontal, as shown, the beam just strikes the top of the building.  a) Find the height of the building and b) the distance the flashlight beam has to travel before it strikes the top of the building. This method is called Triangulation. What do I know? What do I want? Course of action The angle The adjacent side The opposite side USE TANGENT 15

Example A truck driver moves up a straight mountain highway, as shown below. Elevation markers at the beginning and ending points show that he has risen vertically 0.530 km, and the mileage indicator on the truck shows that he has traveled a total distance of 3.00 km during the ascent. Find the angle of incline of the hill. What do I know? What do I want? Course of action The hypotenuse The opposite side The Angle USE INVERSE SINE 16

SI units for Physics SI stands for "System International”. There are 3 fundamental SI units. LENGTH, MASS, and TIME. SI Quantity SI Unit Length Meter Mass Kilogram Time Second Units can be expressed with prefixes attached to make the number larger or smaller. Example: 1 Kilometer – The prefix denotes that the number is actually larger than "1” fundamental units. This is the same as 1000 fundamental units or 1000 meters. 17

Commonly used prefixes in Physics Factor Symbol Tera- (commonly used for computer storage) x 109 T Mega- (used for radio station frequencies and computer storage) x 106 M Kilo- (commonly used for distance and mass, Europe uses the Kilometer instead of the mile on its road signs) x 103 k Centi- (used for small distances) x 10-2 c Milli- (used for small distances or masses) x 10-3 m Micro- (used in electronics) x 10-6 Nano (used in nanotechnology) x 10-9 n 18

Examples The FARAD is the fundamental unit of charge stored on capacitors If a capacitor is labeled 2.5mF(microFarads), how would it be labeled in just Farads? 2.5 x 10-6 F Since the prefix micro ( m ) is 10-6 , add that factor with the base unit. A HERTZ is the fundamental unit used for radio frequency The radio station WEQX 102.7 transmits at a frequency of 102.7 x 106 Hertz. How would it be written in MHz (MegaHertz)? Since the prefix mega ( M ) is 106 , drop that factor and add the prefix to the base unit. 102.7 MHz 19

Dimensional Analysis Dimensional Analysis is a technique used to convert from one unit to another. Remember that the GIVEN UNIT MUST CANCEL OUT (or DIVIDE TO 1). To convert 65 mph to ft/s or m/s.    20

Examples Convert the following: 21

Significant Figures When dealing with measurements, the accuracy of the measurement is specified through the use of significant figures (or digits). When taking a measurement the significant figures depends on the tool used. This term specifies the accuracy of the measurement, which was determined by the tool. The last digit is the one that must be estimated on the ruler. This measurement has three significant figures. 22

Examples Measure the objects below according to rulers shown. 1.36 ± 0.01 cm 2.4 ± 0.1 cm 23

Significant Figures When given a measurement, the number of significant figures can be determined from the number. Examples: 23.2 3 sig fig 1.0001 5 sig fig 0.003 1 sig fig 24

Significant Figures Remember – significant figures only apply to measurements. 25

Accuracy and Precision The difference between accuracy and precision in a set of measurements is illustrated below: 26