QUANITY NAME OF UNIT ABBREVIATION length meter m mass kilogram kg time second s

Transcription

1 Mathematics Review Sheet AP Physics 1 Systems of Units Physics involves an objective description of the world, so measurement is a crucial tool. In measuring, we make use of a group of standard units comprising a system of units. The SI system of units, which is the modern version of the metric system, is the primary system of units that we will use in this class. Units in the SI system include base units, which are defined by standards, and derived units, which are defined in terms of the base units. Some base units are: QUANITY NAME OF UNIT ABBREVIATION length meter m mass kilogram kg time second s Some examples of derived units are: QUANITY NAME OF UNIT ABBREVIATION Area meter squared m 2 Volume meter cubed m 3 speed, velocity meter per second m/s Acceleration meter per second squared m/s 2 Scientific Notation 5.0 x 8 positive powers are used for big numbers. 8 move decimal +8 spots x -6 are negative powers used for small numbers -6 move decimal -6 spots E notation When writing a number with E notation, the basic rule is that the E replaces the "x " in the normal scientific notation form and the exponent that is normally written as a superscript (smaller and raised slightly Like This ) is written as a normal number down on the same level and not raised up. So basically all you need to do to put a number in E notation is take away the "x " and replace it with an E and then write the exponent number next to the E. Examples 6 x -4 = 6E x 6 = 3.405E6-6 x 9 = -6E9 Scientific Notation and the Calculator When you work with numbers that are in scientific notation in a calculator, use the 'E' notation. Do not try to put the numbers in with "x " factors, something always goes wrong with it. It's

2 much easier to work with the E notation and it works well with a calculator. The TI programming calculators have an EE button located on them. When putting scientific E notation numbers in a calculator, press the EE button when it is time to put the "E" in. Other types of calculators will either have the same EE button or have a button labeled "EXP" that you can use to put an "E" in for the "E" notation. Practice with your calculator to get used to the E notation. A number in E notation is the same as a normal number you are used to. You can do the same arithmetic operations on E notation numbers that you can on normal form numbers. For example, = (6x 4 ) + (5 x 3 ) = (6.5x 4 ) Try the E notation on your calculator. 6E4 + 5E3 = Prefixes Prefixes are often used as shorthand for scientific notation. You should be familiar with the SI prefixes. MULTIPLE PREFIX ABBREVIATION MULTIPLE PREFIX ABBREVIATION 12 tera- T -2 centi- c 9 giga- G -3 milli- m 6 mega- M -6 micro- µ 3 kilo- k -9 nano- n 2 hecto- h -12 pico- p 1 deka- da -15 femto- f -1 deci- d The prefixes in bold are the ones most commonly used in this class and should be memorized. Unit Conversions EXAMPLE #1: Convert 5 km to meters. Using the train track method: When you are converting a number from one unit to another be sure to include units for all numbers in your calculation and cancel each unit as you go. Look at your factors to see what you can do. We see that 00 m = 1 km, and we want go to 'm'. We need to get rid of the 'km'. To get rid of the 'km' we will put it on the bottom and divide by it so it will cancel out. So we put the m on top and the km on the bottom so that the km will cancel out. Now you cancel your units. The km s cancel so that you are left with meters. Multiply the numbers through and your final answer is in meters. 5km 00m =5000m 1km

3 Using the prefix substitution method: This is a lot easier! When you see the prefix you just substitute the multiple that is associated with that prefix. Looking at our previous example, 5 km to meters we have kilo meaning 3. Therefore, 5 km = 5 3 meters. You may also be asked to convert metric to English or English to metric. Some important factors are listed in the table to the left. EXAMPLE #2 What if you have to convert a number that has units on the bottom? To get rid of something when it s on the bottom, you have to multiply by it rather than divide (when you have something on the bottom, you want to put it on top when you are converting so it will cancel out). Convert 5 m/s to m/hr. We will use the fact that there are 3600 seconds in 1 hr to make this conversion. EXAMPLE #3 A double conversion. Convert 15 m/s (meters per second) to mph (miles per hour) For this problem you have to do two conversions. First, you have to convert the seconds to hours (which are on the bottom since its m/s to mi/hr). Then you have to convert the m to miles. Let s do it in two steps so it will be easier to see. First we will convert seconds into hours: Now we convert the m into mi

4 Units raised to a power There is a trick to doing conversion problems with units that are squared or cubed (ft 2 or m 3 ). Very few people know how to do this, and it is difficult to understand. Examples are presented here for your reference, though it will be discussed in class to clarify. EXAMPLE #1 Convert square miles to square feet. Notice when executing this conversion, you must square both the number and the unit. EXAMPLE #2 Convert 0 cm 2 to ft 2. Dimensional Analysis Dimensional analysis is a procedure that you can use to check whether an equation is dimensionally correct or incorrect. You expect the two sides of an equation to be equal; thus they must have the same dimensions. To perform dimensional analysis, replace each variable with the type of dimensions it measures, and then algebraically manipulate the dimensions to simplify. If the left hand side and right hand side have the same dimensions, the equation is dimensionally correct. EXAMPLE #1 Show that the equation: v = v 0 + at is dimensionally correct. v = v 0 + at [m/s] = [m/s] + [m/s 2 ][s] [m/s] = [m/s] + [m/s] [m/s] = [m/s] replace quantities with units multiply units add units Therefore the equation is dimensionally correct. Geometry Review

5 Trig functions sin, cos, tan. It is not real important that you know what these are, just that you know how to use them. These functions are used to find the sides and angles and triangles. sin θθ = opposite hypotenuse Finding the unknown side to a triangle: cos θθ = Method 1: Pythagorean Theorem a 2 +b 2 =c 2 EXAMPLE #1 Find side c on the triangle given below. aadjacent hyypotenuse tan θθ = opposite aadjacent Method 2: Using the Trig Functions to find a side NOTE: When using any trig function, your calculator should be set to degree mode. A regular non graphing calculator is usually set to degree mode, but if you use a TI type graphing calculator, you need to switch the mode from radian to degree Any time you replace the batteries in your graphing calculator you need to make this change as well. You will get incorrect answers if you are in the wrong mode. To test it, Try entering (sin 30) if this comes out to 0.5, you are ok. EXAMPLE #2: Suppose you had the following problem: Find side a of the triangle shown below. Look for a trig formula that has all those terms in it and we see the 'tan' formula contains all those items.

6 Graphing Data and theories can often be presented more clearly by graphs than by equations or tables of numbers. Graphs provide more information than can be obtained directly from a table of data. Labeling Axis The independent variable is always on the x-axis and the dependent variable is always on the y-axis. The following conventions are used: y depends on x x is varied by the experimenter; y is measured y as a function of x y versus x Since x is chosen by the experimenter, it usually has even increments. The quantity of time is usually plotted on the x-axis.