Gen. Physics 1 Quarter 1 Module PDF

**Gen. Physics 1** **Quarter 1 -- Module** GENERAL PHYSICS 1 **First Quarter Module** **Prepared by EDGAR M. UBALDE, LPT** The module consists of all the Most Essential Learning Competencies which was determined by the Department of Education. The First Quarter period is devoted to the first b...

**Gen. Physics 1** **Quarter 1 -- Module** GENERAL PHYSICS 1 **First Quarter Module** **Prepared by EDGAR M. UBALDE, LPT** The module consists of all the Most Essential Learning Competencies which was determined by the Department of Education. The First Quarter period is devoted to the first branch of classical physics. Mechanics is concerned with the motions of physical objects and the relationships between force, matter, and motion. **For 9 weeks you are expected to:** **Week 2** - Solve measurement problems involving conversion of units, expression of measurements in scientific notation - Differentiate accuracy from precision - Differentiate random errors from systematic errors - Estimate errors from multiple measurements of a physical quantity using variance **Week 3** - Differentiate vector and scalar quantities - Perform addition of vectors - Rewrite a vector in component form **Week 4** - Convert a verbal description of a physical situation involving uniform acceleration in one dimension into a mathematical description - Interpret displacement and velocity, respectively, as areas under velocity vs. time and acceleration vs. time curves - Interpret velocity and acceleration, respectively, as slopes of position vs. time and velocity vs. time curves **Week 5** - Describe motion using the concept of relative velocities in 1D and 2D - Deduce the consequences of the independence of vertical and horizontal components of projectile motion - Calculate range, time of flight, and maximum heights of projectiles - Infer quantities associated with circular motion such as tangential velocity, centripetal acceleration, tangential acceleration, radius of curvature **Week 6** - Define inertial frames of reference - Identify action -reaction pairs - Draw free -body diagrams - Apply Newton's 1st law to obtain quantitative and qualitative conclusions about the contact and noncontact forces acting on a body in equilibrium - Apply Newton's 2nd law and kinematics to obtain quantitative and qualitative conclusions about the velocity and acceleration of one or more bodies, and the contact and noncontact forces acting on one or more bodies **Week 7** - Differentiate the properties of static friction and kinetic friction - Calculate the dot or scalar product of vectors - Determine the work done by a force acting on a system - Define work as a scalar or dot product of force and displacement - Interpret the work done by a force in one dimension as an area under a Force vs. Position curve **Week 8** - Relate the gravitational potential energy of a system or object to the configuration of the system - Relate the elastic potential energy of a system or object to the configuration of the system - Explain the properties and the effects of conservative forces - Use potential energy diagrams to infer force; stable, unstable, and neutral equilibrium; and turning points **Week 9** - Differentiate center of mass and geometric center - Relate the motion of center of mass of a system to the momentum and net external force acting on the system - Relate the momentum, impulse, force, and time of contact in a system - Compare and contrast elastic and inelastic collisions - Apply the concept of restitution coefficient in collisions **PERFORMANCE TASKS** 1. Solve for unknown quantities in equations involving one-dimensional uniformly accelerated motion, including free fall motion 2. Solve problems involving one-dimensional motion with constant acceleration in contexts such as, but not limited to, the "tail-gating phenomenon", pursuit, rocket launch, and free fall problems 3. Solve problems involving two dimensional motion in contexts such as, but not limited to ledge jumping, movie stunts, basketball, safe locations during firework displays, and Ferris wheels 4. Solve problems using Newton's Laws of motion in contexts such as, but not limited to, ropes and pulleys, the design of mobile sculptures, transport of loads on conveyor belts, force needed to move stalled vehicles, determination of safe driving speeds on banked curved roads 5. Solve problems involving center of mass, impulse, and momentum in contexts such as, but not limited to, rocket motion, pingpong and vehicle collisions. **INTRODUCTION TO PHYSICS** +-----------------------------------+-----------------------------------+ | | **Physics is the science that | | | deals with the structure of | | | matter and the interactions | | | between the fundamental | | | constituents of the observable | | | universe.** In the broadest | | | sense, physics (from the Greek | | | physikos) is concerned with all | | | aspects of nature on both the | | | macroscopic and sub microscopic | | | levels. | +-----------------------------------+-----------------------------------+ | | **Physics as a Specialized | | | Subject** | | | | | | Learners will go through various | | | concepts that deals with the | | | combination of matter and energy. | | | | | | Energy may manifest itself in | | | many forms, such as light, | | | motion, gravity, radiation, | | | electricity, and others. Matter, | | | on the other hand, covers any and | | | every physical manifestation, | | | from the smallest particles such | | | as atoms and sub-atomic | | | particles, further into larger | | | physical groups such as stars, | | | universes, and galaxies. | | | | | | **Physics can also be described | | | as the science dealing with | | | physical quantities.** In this | | | regard, physics is widely | | | considered to be the most | | | fundamental (and important) of | | | all the natural sciences. **After | | | all, physics pertains to the | | | quantification of almost all | | | matter that exists in this | | | world.** It is any aspect of | | | nature that can be expressed, | | | measured or calculated in | | | specific terms. | | | | | | **In this regard, physics and | | | mathematics are closely related | | | to one another. It could be said | | | that mathematics is the language | | | of physics.** That is another way | | | of better understanding what | | | physics is. Numerical values, | | | units of measurement, and similar | | | concepts are all mathematical in | | | nature, and are used to describe | | | physical quantities in the most | | | accurate and precise manner. | +-----------------------------------+-----------------------------------+ | | | +-----------------------------------+-----------------------------------+ **MEASUREMENTS** **MELC** Solve measurement problems involving conversion of units, expression of measurements in scientific notation +-----------------------------------+-----------------------------------+ | **WHAT IS IT** | There is a lot of mathematics | | | problem solving in our subject. | |  | though calculus. We are not going | | | through that difficult path, we | | | will take a much easier one- that | | | is through algebra. To begin our | | | study of Physics, we must go back | | | to the basics of algebra. | | | | | | **RULES FOR COUNTING SIGNIFICANT | | | NUMBERS** | | | | | | **All non-zero digits are | | | significant**. | | | | | | [Example] : 12345 =\> | | | 5 sf ; 21.4327 =\> 6 sf | | | | | | **All zero digits occurring | | | between non-zero digits are | | | significant.** | | | | | | [Example]: 123450.05 | | | =\> 8 sf | | | | | | **All non-zero digit to the right | | | of the decimal point are | | | significant figure**. | | | [Exampl]e: 0.08 =\> 1 | | | sf; 0.008 =\> 1 sf | | | | | | **Trailing zeros after a decimal | | | point are significant numbers. | | | Since this shows accuracy.** | | | | | | [Example:] 12345 =\> | | | 5 sf; 12345.0 =\>? 12345.00 =\>? | | | | | | **Significant figure is | | | conserved. Significant figures | | | should not change even if the | | | unit of measurement is changed.** | | | | | | [Example]: length is | | | equivalent to 85 mm =\> 2 sf | | | | | | **SCIENTIFIC NOTATION** | | | | | | A short hand method of **writing | | | numbers using the power of 10**. | | | It was developed in order to | | | easily represent numbers that are | | | either very large or very small. | | | | | | **When writing scientific | | | notation, the decimal point is | | | always moved after the 1st | | | non-zero number. Count the number | | | of times the decimal point is | | | moved and use this as the power | | | of 10.** | | | | | | Big numbers \>1 have positive | | | exponent while Small numbers \ 1.027500456x10⁶ ; | | | 0.007543 =\> 7.543x10⁻⁴ | | | | | | **CALCULATIONS WITH SCIENTIFIC | | | NOTATION** | | | | | | **Adding** (with same power of | | | 10), add the numbers. Keep the | | | power of 10**.** | | | [Example]: 2x10^3^ + | | | 3x10^3^ = 5x10^3^ | | | | | | **Subtracting** (with the same | | | power of 10), subtract numbers. | | | Keep power of 10 | | | [Example]: 3x10^3^ -- | | | 2x10^3^ = 1x10^3^ | | | | | | **Multiplying** multiply numbers. | | | Add powers of 10 | | | | | | [Example]: 2x10^6^ \* | | | 3x10^3^ = 6x10⁹ | | | | | | **Dividing: divide the numbers & | | | subtract the powers of 10** | | | | | | [Example]: 2x10^6^ / | | | 3x10^3^ = 0.67x10^3^ | | | | | | **Taking it to a power:** take | | | the number to the power. Multiply | | | the power of 10 by the power. | | | | | | [Example]: | | | (2x10^3^)^3^ = 8x10^9^ | | | | | | **CALCULATIONS WITH EXPONENTS** | | | | | | **Power of "1:** anything to the | | | power of 1 is itself. | | | [Example]: 25^1^ | | | =**25** | | | | | | **When Multiplying exponents**: | | | Add the powers | | | [Example:] | | | 3^4^\*3^5^=**3^9^** | | | | | | **When Dividing exponents:** | | | Subtract the powers | | | [Example:] | | | 3^4^-3^2^=**3^2^** | | | | | | **Power of a power:** Multiply | | | the powers [Example:] | | | (3^2^)^4^=**3^8^** | | | | | | **Negative Powers:** Put the | | | number on the opposite side of | | | the fraction and power becomes | | | positive. [Example:] | | | 3^-2^**=1/3^2^** ^;^ | | | 8^-2^=**1/8^2^** | +-----------------------------------+-----------------------------------+ | **WHAT I KNOW** | **Activity 1.1 Convert the | | | following** | | C:\\Users\\DEPED\\Desktop\\ADM | | | specs\\ADM cover and icons\\ADM | 1\. Convert 6721 millimetres to | | Icons\\What I Know.jpg | meters. | | | | | | 2\. If 2 mL of liquid weighs 4 | | | g, its density is | | | | | | 3\. If the density of a | | | substance is 8 g/mL, what | | | volume would 40 g of the | | | substance occupy? | | | | | | 4\. Convert 300[⁰]{.math | | |.inline}C to 0ºF | | | | | | 5\. How many cubic centimetre | | | are there in a cubic meter? | +-----------------------------------+-----------------------------------+ | **WHAT'S MORE** | **Determination of Quantity in | | | terms of a Chosen Unit**. The | |  | 1**. Fundamental Unit-** upon | | | which other units are derived. | | | | | | **[Examples]**- | | | Meter, Kilogram, Second | | | | | | 2. **Derived Units** -- units | | | derived from simple | | | combination of two or more | | | fundamental units. | | | | | | **UNIT OF MEASUREMENT** | | | | | | **SI Units (Systeme International | | | d\'Unites)** | | | | | | ***Table 1.** SI Base Units* | | | | | | **Quantity** **Name** | | | **Symbol** | | | --------------------- --------- | | | - ------------ | | | Length Meter | | | M | | | Mass Kilogram | | | Kg | | | Time Second | | | S | | | Electric current Ampere | | | A | | | Temperature Kelvin | | | K | | | Amount of substance Mole | | | Mol | | | Luminous intensity Candela | | | Cd | | | | | | **Metric Prefixes** | | | | | | ***[Table | | | 2](https://www.texasgateway.org/r | | | esource/13-language-physics-physi | | | cal-quantities-and-units#Table_01 | | | _03_Metrics) **Metric | | | Prefixes and symbols used to | | | denote the different various | | | factors of 10 in the metric | | | system* | | | | | | **Prefix** **EXAMPLE** | | | | | | | | | -------------- ------------- -- | | | ---------- ------------ --------- | | | --- ------------ ---------------- | | | --------------------- | | | **Symbol** ** | | | Value** **Name** **Symbol* | | | * **Value** **Description** | | | **Exa** E 10 | | | ^18^ Exameter Em | | | 10^18^ m Distance light t | | | ravels in a century | | | **Peta** P 10 | | | ^15^ Petasecond Ps | | | 10^15^ s 30 million years | | | **Tera** T 10 | | | ^12^ Terawatt TW | | | 10^12^ W Powerful laser o | | | utput | | | **Giga** G 10 | | | ^9^ Gigahertz GHz | | | 10^9^ Hz A microwave freq | | | uency | | | **Mega** M 10 | | | ^6^ Megacurie MCi | | | 10^6^ Ci High radioactivi | | | ty | | | **Kilo** K 10 | | | ^3^ Kilometer Km | | | 10^3^ m About 6/10 mile | | | **hector** H 10 | | | ^2^ Hectoliter hL | | | 10^2^ L 26 gallons | | | **Deka** Da 10 | | | ^1^ Dekagram Dag | | | 10^1^ g Teaspoon of butt | | | er | | | **\_\_\_\_** \_\_\_\_ 10 | | | ^0^ (=1) | | | | | | **Deci** D 10 | | | ^--1^ Deciliter dL | | | 10^--1^ L Less than half a | | | soda | | | **Centi** C 10 | | | ^--2^ Centimeter Cm | | | 10^--2^ m Fingertip thickn | | | ess | | | **Mili** M 10 | | | ^--3^ Millimeter Mm | | | 10^--3^ m Flea at its shou | | | lder | | | **Micro** µ 10 | | | ^--6^ Micrometer µm | | | 10^--6^ m Detail in micros | | | cope | | | **Nano** N 10 | | | ^--9^ Nanogram Ng | | | 10^--9^ g Small speck of d | | | ust | | | **Pico** P 10 | | | ^--12^ Pico farad pF | | | 10^--12^ F Small capacitor | | | in radio | | | **Femto** F 10 | | | ^--15^ Femtometer Fm | | | 10^--15^ m Size of a proton | | | **Atto** A 10 | | | ^--18^ Attosecond As | | | 10^--18^ s Time light takes | | | to cross an atom | | | | | | The metric system is convenient | | | because conversions between | | | metric units can be done simply | | | by moving the decimal place of a | | | number. This is because the | | | metric prefixes are sequential | | | powers of 10. There are 100 | | | centimeters in a meter, 1000 | | | meters in a kilometer, and so on. | | | In nonmetric systems, such as | | | U.S. customary units, the | | | relationships are less | | | simple---there are 12 inches in a | | | foot, 5,280 feet in a mile, 4 | | | quarts in a gallon, and so on. | | | | | | **Unit Conversion and Dimensional | | | Analysis** | | | | | | A **conversion factor** relating | | | meters to kilometers. | | | A **conversion factor** is a | | | ratio expressing how many of one | | | unit are equal to another unit. | | | **A conversion factor is simply a | | | fraction which equals 1.** | | | | | | You can multiply any number by 1 | | | and get the same value. When you | | | multiply a number by a conversion | | | factor, you are simply | | | multiplying it by one. | | | | | | For example, the following are | | | conversion factors: | | | | | | ***1 foot/12 inches = 1 to | | | convert inches to feet, 1 | | | meter/100 centimeters*** | | | | | | ***= 1 to convert centimeters to | | | meters,*** | | | | | | ***1 minute/60 seconds = 1 to | | | convert seconds to minutes*** | | | | | | In this case, we know that there | | | are 1,000 meters in 1 kilometer. | | | | | | We will write the units that we | | | have and then multiply them by | | | the conversion factor. | +-----------------------------------+-----------------------------------+ | **ASSESSMENT**C:\\Users\\DEPED\\D | **Activity 1.2** | | esktop\\ADM | | | specs\\ADM cover and icons\\ADM | A. **Convert the given | | Icons\\Tayahin.jpg | quantities:** | | | | | **WHAT I HAVE LEARNED** | | | | | |  | | | | | | | 3. 1.2 GV to Volt 4. 4.6 ms to | | | second | | | | | | | | | | | | 5. 450 K to ^0^Fahrenheit | | | | | | | | | | | | B. **Express the following | | | numbers in scientific | | | notation.** | | | | | | | | | | | | 1. 98 2. 0.0026 3. 0.0000401 4. | | | 643.9 5. 816000 | | | | | | | | | | | | C. **Transform the following | | | scientific notation to | | | standard notation** | | | | | | | | | | | | 1. 6.455 x 10^4^ 2. 3.1 x | | | 10^-6^ 3. 5.00 x 10^-2^ 4. | | | 7.2 x 10^3^ 5. 9 x 10-^5^ | | | | | | **Physical quantities are unit | | | that describes the size of the | | | quantity**. There are number that | | | gives us the count of times the | | | unit is contained in the quantity | | | being measured. | | | | | | Physical Quantities are | | | classified as fundamental and | | | derived quantities. | | | | | | **Fundamental Quantities are the | | | simplest form.** | | | | | | **Derived Quantities are | | | combination of fundamental | | | Quantities.** | | | | | | Systems of measurement are Metric | | | System of System International | | | (SI) and English System or | | | British System of measurement. | | | | | | **Conversion of unit common | | | method used is the factor-label | | | method.** | | | | | | **Scientific Notation is a | | | convenient way of writing very | | | small or very large numbers**. | | | | | | To write in scientific notation, | | | follow the form **N x 10^a^**, | | | where N is a number between 1 and | | | 10, but not 10 itself, a is an | | | integer (positive or negative | | | number) | +-----------------------------------+-----------------------------------+ | **PRECISION & ACCURACY** | | | | | | **MELC** Differentiate accuracy | | | from precision | | +-----------------------------------+-----------------------------------+ | **WHAT IS IT** | **Accuracy** is the degree of | | | conformity and correctness of | | C:\\Users\\DEPED\\Desktop\\ADM | something when compared to the | | specs\\ADM cover and icons\\ADM | true or absolute value. | | Icons\\suriin.jpg | | | | **Precision** refers to the | | | closeness of two or more | | | measurements to each other. | | | | | |  | +-----------------------------------+-----------------------------------+ | **WHAT'S MORE** | Precision and accuracy are two | | | ways that scientists think about | | C:\\Users\\DEPED\\Desktop\\ADM | error. **Accuracy refers to how | | specs\\ADM cover and icons\\ADM | close a measurement is to the | | Icons\\whats more.jpg | true or accepted value. Precision | | | refers to how close measurements | | | of the same item are to each | | | other. Precision is independent | | | of accuracy.** | | | | | | That means it is possible to be | | | very precise but not very | | | accurate, and it is also possible | | | to be accurate without being | | | precise. **The best quality | | | scientific observations are both | | | accurate and precise.** | | | | | | A classic way of demonstrating | | | the difference between precision | | | and accuracy is with a dartboard. | | | Think of the bulls-eye (center) | | | of a dartboard as the true value. | | | The closer darts land to the | | | bulls-eye, the more accurate they | | | are. If the darts are neither | | | close to the bulls-eye, nor close | | | to each other, there is **neither | | | accuracy, nor precision**. If all | | | of the darts land very close | | | together, but far from the | | | bulls-eye, there is **precision, | | | but not accuracy.** | | | | | | If the darts are all about an | | | equal distance from and spaced | | | equally around the bulls-eye | | | there is mathematical accuracy | | | because the average of the darts | | | is in the bulls-eye. This | | | represents data that is accurate, | | | but not precise However, if you | | | were actually playing darts this | | | would not count as a bulls-eye! | | | | | | **If the darts land close to the | | | bulls-eye and close together, | | | there is both accuracy and | | | precision**. | | | | | | **ERRORS** | +-----------------------------------+-----------------------------------+ | | **Errors are a measure of the | | | lack of certainty in a value**. | | | All experimental | | | | | | uncertainty is due to either | | | random errors or systematic | | | errors. | | | | | | **Random errors are statistical | | | fluctuations (in either | | | direction) in the measured data | | | due to the precision limitations | | | of the measurement device.** | | | | | | Random errors usually result from | | | the experimenter\'s inability to | | | take the same measurement in | | | exactly the same way to get exact | | | the same number. | | | | | | **Systematic errors, by contrast, | | | are reproducible inaccuracies | | | that are consistently in the same | | | direction. Systematic errors are | | | often due to a problem which | | | persists throughout the entire | | | experiment.** | | | | | | Note that systematic and random | | | errors refer to problems | | | associated with making | | | measurements. Mistakes made in | | | the calculations or in reading | | | the instruments are not | | | considered in error analysis. It | | | is assumed that the experimenter | | | are careful and competent! | +-----------------------------------+-----------------------------------+ | | | +-----------------------------------+-----------------------------------+ | **WHAT I KNOW** | **Random errors in experimental | | | measurements are caused by | |  | measuring instruments or in the | | | environmental conditions. | | | **Typical causes of systematic | | | error include observational | | | error, imperfect instrument | | | calibration, and environmental | | | interference.** | | | | | | For example: Forgetting to tare | | | or zero a balance produces mass | | | measurements that are always | | | \"off\" by the same amount. | | | | | | **How to Minimize Experimental | | | errors** | +-----------------------------------+-----------------------------------+ | **WHAT'S MORE** | +---------+---------+---------+ | | | | **Type | **Examp | **How | | | C:\\Users\\DEPED\\Desktop\\ADM | | of | le** | to | | | specs\\ADM cover and icons\\ADM | | Error** | | Minimiz | | | Icons\\whats more.jpg | | | | e** | | | | +=========+=========+=========+ | | | | **Rando | You | Take | | | | | m | measure | more | | | | | Error** | the | data. | | | | | | mass of | Random | | | | | | a ring | errors | | | | | | three | can be | | | | | | times | evaluat | | | | | | using | ed | | | | | | the | through | | | | | | same | statist | | | | | | balance | ical | | | | | | and get | analysi | | | | | | slightl | s | | | | | | y | and can | | | | | | differe | be | | | | | | nt | reduced | | | | | | values. | by | | | | | | 17.46 | averagi | | | | | | g.,17.4 | ng | | | | | | 2 | over a | | | | | | g., | large | | | | | | 17.44 | number | | | | | | | of | | | | | | | observa | | | | | | | tions | | | | +---------+---------+---------+ | | | | **Syste | The | Systema | | | | | matic | cloth | tic | | | | | Error** | tape | errors | | | | | | measure | are | | | | | | that | difficu | | | | | | you use | lt | | | | | | to | to | | | | | | measure | detect | | | | | | the | and | | | | | | length | cannot | | | | | | of an | be | | | | | | object | analyse | | | | | | had | d | | | | | | been | statist | | | | | | stretch | ically, | | | | | | ed | because | | | | | | from | all of | | | | | | years | the | | | | | | of use. | data is | | | | | | (As a | off in | | | | | | result, | the | | | | | | all of | same | | | | | | your | directi | | | | | | measure | on | | | | | | ments | (too | | | | | | in | high or | | | | | | length | too | | | | | | were | low). | | | | | | smaller | Spottin | | | | | | ) | g | | | | | | | and | | | | | | The | correct | | | | | | electro | ing | | | | | | nic | takes a | | | | | | scale | lot of | | | | | | you use | care. | | | | | | reads | | | | | | | 0.05 g | | | | | | | higher | | | | | | | for all | | | | | | | mass | | | | | | | measure | | | | | | | ments. | | | | | | | (Improp | | | | | | | erly | | | | | | | tared | | | | | | | through | | | | | | | out | | | | | | | the | | | | | | | experim | | | | | | | ent) | | | | | +---------+---------+---------+ | +-----------------------------------+-----------------------------------+ | | **Estimating Errors** | | | | | | We would like you to think about | | | the measurements and to form some | | | opinion as to how to estimate the | | | error. There will possibly be | | | several acceptable methods. There | | | may be no "best" method. | | | Sometimes "best" is a matter of | | | opinion. | | | | | | **When attempting to estimate the | | | error of a measurement, it is | | | often important to determine | | | whether the sources of error are | | | systematic or random**. A single | | | measurement may have multiple | | | error sources, and these may be | | | mixed systematic and random | | | errors. | | | | | | To identify a random error, the | | | measurement must be repeated a | | | small number of times. **If the | | | observed value changes apparently | | | randomly with each repeated | | | measurement, then there is | | | probably a random error**. The | | | random error is often quantified | | | by the standard deviation of the | | | measurements. Note that more | | | measurements produce a more | | | precise measure of the random | | | error. | | | | | | To detect a systematic error is | | | more difficult. The method and | | | apparatus should be carefully | | | analysed. Assumptions should be | | | checked**. If possible, a | | | measurement of the same quantity, | | | but by a different method, may | | | reveal the existence of a | | | systematic error. A systematic | | | error may be specific to the | | | experimenter.** Having the | | | measurement repeated by a variety | | | of experimenters would test this. | | | | | | **Variance (σ^2^) in statistics | | | is a measurement of the spread | | | between numbers in a data set**. | | | That is, it measures how far each | | | number in the set is from the | | | mean and therefore from every | | | other number in the set. | | | | | | **[Example:]** | | | | | | **Precision is often reported | | | quantitatively by using relative | | | or fractional uncertainty:** | | | | | | Relative Uncertainty = | | | uncertainty/measured quantity | | | | | | Example: m = 75.5 ± 0.5 g | | | | | | It has a fractional uncertainty | | | of: [\$\\frac{0.5\\ g}{75.5\\ | | | g}\$]{.math.inline} = 0.006 = | | | 0.7%. | | | | | | **Accuracy is often reported | | | quantitatively by using relative | | | error** | | | | | | Relative Error = | | | [\$\\frac{measured\\ value\\ - \\ | | | expected\\ | | | value}{\\text{expected\\ | | | value}}\$]{.math.inline} | | | | | | If the expected value for m is | | | 80.0 g, then the relative error | | | is: | | | | | | [\$\\text{\\ \\ \\ \\ \\ \\ \\ \\ | | | \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ | | | \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ | | | \\ \\ \\ \\ \\ \\ \\ \\ \\ | | | }\\frac{75.5\\ - \\ | | | 80.0}{80.0}\$]{.math.inline}= | | | −0.056 = −5.6% | | | | | | **Note:** The minus sign | | | indicates that the measured value | | | is less than the expected value. | | | When analysing experimental data, | | | it is important that you | | | understand the difference between | | | precision and accuracy. | | | | | | **Precision indicates the quality | | | of the measurement, without any | | | guarantee that the measurement is | | | \"correct.\" Accuracy, on the | | | other hand, assumes that there is | | | an ideal value, and tells how far | | | your answer is from that ideal, | | | \"right\" answer.** These | | | concepts are directly related to | | | random and systematic measurement | | | errors. | | | | | | **VECTORS** | +-----------------------------------+-----------------------------------+ | **MELC** Differentiate vector and | | | scalar quantities | | | | | | Perform addition of vectors | | +-----------------------------------+-----------------------------------+ | **WHAT I KNOW** | We will encounter a lot of things | | | related to vectors and scalar | |  | Mathematicians and scientists | | | call a quantity which **depends | | **WHAT'S MORE** | on direction** a vector quantity. | | | A quantity which **does not | | C:\\Users\\DEPED\\Desktop\\ADM | depend on direction** is called a | | specs\\ADM cover and icons\\ADM | scalar quantity. | | Icons\\whats more.jpg | | | | **SCALAR** | | | | | | Only has magnitude (size) which | | | includes ***distance, speed, | | | time, temperature, mass, length, | | | area, volume, density, charge, | | | pressure, energy, work and | | | power**.* | | | | | | **VECTOR** | | | | | | A geometric representations of | | | magnitude and direction and can | | | be expressed as arrows in two or | | | three dimensions that includes | | | ***displacement, velocity, | | | acceleration, momentum, force, | | | lift, drag, thrust and weight.*** | | | | | | **What set vectors apart from | | | scalar quantities is the arrow or | | | the direction to which physical | | | entity is applied for example, | | | force. A force has a magnitude | | | measured in grams or kilograms | | | and it also has direction**. | | | | | | You may encounter a single vector | | | acting on a single point or a | | | multitude of vectors acting | | | simultaneously on a given point. | | | Vectors can go on the same | | | direction concurrently or go | | | against each other. Either way | | | you may be ask to derive the | | | resultant vector | | | | | | **ADDITION OF VECTORS** This is a | | | vector: | | | | | |  | | | | | | A vector has **magnitude** (size) | | | and **direction**: | | | | | | The length of the line shows its | | | magnitude and the arrowhead | | | points in the direction. We can | | | add two vectors by joining them | | | head-to-tail: | | | | | | And it doesn\'t matter which | | | order we add them, we get the | | | same result: | | | | | | vector add b+a | | | | | | ### **[Example:] A pl | | | ane is flying along, pointing nor | | | th, but there is a wind coming fr | | | om the North-West.** The two vect | | | ors (the velocity caused by the p | | | ropeller, and the velocity of the | | | wind) result in a slightly slowe | | | r ground speed heading a little E | | | ast of North. If you watched the | | | plane from the ground it would se | | | em to be slipping sideways a litt | | | le. | | | | | |  | | | | | | Have you ever seen that happen? | | | Maybe you have seen birds | | | struggling against a strong wind | | | that seem to fly sideways. | | | Vectors help explain that. | | | | | | [Velocity](https://www.mathsisfun | | |.com/measure/metric-speed.html),  | | | [acceleration](https://www.mathsi | | | sfun.com/measure/metric-accelerat | | | ion.html), [force](https://www.ma | | | thsisfun.com/physics/force.html)  | | | and | | | many other things are vectors. | | | | | | SUBTRACTION OF VECTORS | | | ---------------------- | | | | | | We can also subtract one vector | | | from another: | | | | | | - first we reverse the | | | direction of the vector we | | | want to subtract, | | | | | | - then add them as usual: | | | | | | vector subtract a-b = a + (-b)\ | | | **a** − **b** | | | | | | A vector is often written | | | in **bold**, like **a** or **b**. | | | | | | +---------+---------+---------+ | | | | A |   | | | | | | vector | | | | | | | can | | | | | | | also be | | | | | | | written | | | | | | | as the | | | | | | | letters | | | | | | | of its | | | | | | | head | | | | | | | and | | | | | | | tail | | | | | | | with an | | | | | | | arrow | | | | | | | above | | | | | | | it, | | | | | | | like | | | | | | | this: | | | | | | | | | | | | | |  | | | | | | +---------+---------+---------+ | | | | | | Now \... how do we do the | | | calculations? The most common way | | | is to first break up vectors into | | | x and y parts, like this: | | | | | | vector xy components | | | | | | The vector **a** is broken up | | | into the two | | | vectors **a~x~** and **a~y~** | | | | | | Adding Vectors | | | -------------- | | | | | | We can then add vectors | | | by **adding the x | | | parts** and **adding the y | | | parts**: | | | | | |  | | | | | | The vector (8, 13) and the vector | | | (26, 7) add up to the vector (34, | | | 20) | | | | | | ### **Example: add the vectors ** | | | a** = (8, 13) and **b** = (26, 7) | | | ** | | | | | | **c** = **a** + **b** | | | | | | **c** = (8, 13) + (26, 7) = | | | (8+26, 13+7) = (34, 20) | | | | | | **Subtracting Vectors** | | | | | | To subtract, first reverse the | | | vector we want to subtract, then | | | add. | | | | | | **[Example]: | | | subtract** k = (4, 5) from v = | | | (12, 2) **a = v + −k** | | | | | | a = (12, 2) + −(4, 5) = (12, 2) + | | | (−4, −5) = (12−4, 2−5) = (8, −3) | | | | | | **Magnitude of a Vector** is | | | shown by two vertical bars on | | | either side of the | | | vector:\|**a**\| OR it can be | | | written with double vertical bars | | | (so as not to confuse it with | | | absolute value): \|\|**a**\|\| | | | | | | We use [**Pythagoras\' | | | theorem**](https://www.mathsisfun | | |.com/pythagoras.html) to | | | calculate it: \|**a**\| = **√( | | | x^2^ + y^2^ )** | | | | | | ### **[Example:] What | | | is the magnitude of the vector * | | | *b** = (6, 8)?** | | | | | | \|**b**\| = √( 6^2^ + 8^2^) = √( | | | 36+64) = √100 = 10 | +-----------------------------------+-----------------------------------+ | **MELC** Rewrite a vector in | | | component form | | +-----------------------------------+-----------------------------------+ | **WHAT I KNOW** | **In situations in which vectors | | | are directed at angles to the | | C:\\Users\\DEPED\\Desktop\\ADM | customary coordinate axes, a | | specs\\ADM cover and icons\\ADM | useful mathematical trick will be | | Icons\\What I Know.jpg | employed to transform the vector | | | into two parts with each part | | **WHAT IT IS** | being directed along the | | | coordinate axes.** For example, a | |  | westward part. A vector that is | | | directed upward and rightward can | | **WHAT'S MORE** | be thought of as having two parts | | | - an upward part and a rightward | | C:\\Users\\DEPED\\Desktop\\ADM | part. | | specs\\ADM cover and icons\\ADM | | | Icons\\whats more.jpg |   | | | | | | **Any vector directed in two | | | dimensions can be thought of as | | | having an influence in two | | | different directions. That is, it | | | can be thought of as having two | | | parts. Each part of a | | | two-dimensional vector is known | | | as a component**. The components | | | of a vector depict the influence | | | of that vector in a given | | | direction. The combined influence | | | of the two components is | | | equivalent to the influence of | | | the single two-dimensional | | | vector. The single | | | two-dimensional vector could be | | | replaced by the two components. | | | | | | **Angled Vectors Have Two | | | Components** | | | | | | If Fido's dog chain is stretched | | | upward and rightward and pulled | | | tight by his master, then the | | | tension force in the chain has | | | two components - an upward | | | component and a rightward | | | component. | | | | | | To Fido, the influence of the | | | chain on his body is equivalent | | | to the influence of two chains on | | | his body - one pulling upward and | | | the other pulling rightward. | | | | | | If the | | | single chain were replaced by two | | | chains. With each chain having | | | the magnitude and direction of | | | the components, then Fido would | | | not know the difference. | | | | | | This is not because Fido | | | is dumb (a quick glance at his | | | picture reveals that he is | | | certainly not that), but rather | | | because the combined influence of | | | the two components is equivalent | | | to the influence of the single | | | two-dimensional vector. | | | | | |   | | | | | | Consider a picture that is hung | | | to a wall by means of two wires | | | that are stretched vertically and | | | horizontally. Each wire exerts a | | | tension force upon the picture to | | | support its weight. Since each | | | wire is stretched in two | | | dimensions (both vertically and | | | horizontally), the tension force | | | of each wire has two components - | | | a vertical component and a | | | horizontal component. Focusing on | | | the wire on the left, we could | | | say that the wire has a leftward | | | and an upward component. | | | | | | This is to say that the wire on | | | the left could be replaced by two | | | wires, one pulling leftward and | | | the other pulling upward. If the | | | single wire were replaced by two | | | wires (each one having the | | | magnitude and direction of the | | | components), then there would be | | | no effect upon the stability of | | | the picture. The combined | | | influence of the two components | | | is equivalent to the influence of | | | the single two-dimensional | | | vector. | | | | | | **Trigonometric Method of Vector | | | Resolution** | | | | | | Trigonometric functions relate | | | the ratio of the lengths of the | | | sides of a right triangle to the | | | measure of an acute angle within | | | the right triangle. The | | | trigonometric method of vector | | | resolution involves using | | | trigonometric functions to | | | determine the components of the | | | vector.  We will use | | | trigonometric functions to | | | determine the components of a | | | single vector.  As such, | | | trigonometric functions can be | | | used to determine the length of | | | the sides of a right triangle if | | | an angle measure and the length | | | of one side are known. | | | | | | **The method of employing | | | trigonometric functions to | | | determine the components of a | | | vector are as follows:** | | | | | | 1\. Construct a rough sketch (no | | | scale needed) of the vector in | | | the indicated direction. Label | | | its magnitude and the angle | | | that it makes with the | | | horizontal. | | | | | | 2\. Draw a rectangle about the | | | vector such that the vector is | | | the diagonal of the rectangle. | | | Beginning at the tail of the | | | vector, sketch vertical and | | | horizontal lines. Then sketch | | | horizontal and vertical lines | | | at the head of the vector. The | | | sketched lines will meet to | | | form a rectangle. | | | | | | 3\. Draw the components of the | | | vector. The components are | | | the sides of the rectangle. The | | | tail of each component begins | | | at the tail of the vector and | | | stretches along the axes to the | | | nearest corner of the | | | rectangle. Be sure to place | | | arrowheads on these components | | | to indicate their direction | | | (up, down, left, right). | | | | | | **COSINE-SINE LAW** | | | | | | When you have two vectors that | | | are not at right angles to each | | | other, you apply the **Cosine-** | | | | | | **Cosine Law is used to find the | | | magnitude.** | | | | | | [Example] | | | | | |  | | | | | | **Sine Law is used to find the | | | direction** | | | | | | [Example] | | | | | |  | +-----------------------------------+-----------------------------------+ | **ASSESSMENT** | **Activity 1.4. Answer the | | | following.** | | C:\\Users\\DEPED\\Desktop\\ADM | | | specs\\ADM cover and icons\\ADM | 1. Find the displacement of an | | Icons\\Tayahin.jpg | airplane that flies 340 km | | | \[W\[\], then 120 km \[S\] | | | and then 220 km \[E\] in 3.00 | | | hours. Then find the plane's | | | average speed and average | | | velocity. (Use Pythagorean | | | Theorem and Trigonometry). | | | | | | 2. A person walks 5.8 km \[N\], | | | 4.0 km \[E\] and finally 3.0 | | | km \[S\] in 2.8 hours. Find | | | the person's resultant | | | displacement and average | | | velocity. (Use Pythagorean | | | Theorem and Trigonometry) | +-----------------------------------+-----------------------------------+ | **ONE DIMENSION KINEMATICS** | | | | | | **MELC** Convert a verbal | | | description of a physical | | | situation involving uniform | | | acceleration in one dimension | | | into a mathematical description. | | +-----------------------------------+-----------------------------------+ | **WHAT IT IS** | **Kinetics** is the study of | | | forces that cause motion while | |  | | | | Whatever speed is, it involves | | | both distance and time. | | | \"Faster\" means either | | | \"farther\" (greater distance) or | | | \"sooner\" (less time). Doubling | | | one\'s speed would mean doubling | | | one\'s distance traveled in a | | | given amount of time. Doubling | | | one\'s speed would also mean | | | halving the time required to | | | travel a given distance. The | | | symbol *v* is used for speed | | | because of the association | | | between speed and velocity, | | | | | | **Speed is directly proportional | | | to distance when time is constant | | | *v* ∝ *s* **(*t* constant) | | | | | | **Speed is inversely proportional | | | to time when distance is | | | constant** | | | | | | ***v* ∝ ^1/^*t* **(*s* constant) | | | | | | Combining these two rules | | | together gives the definition of | | | speed in symbolic form. | | | | | | ***v* =  ** ***[s]{.underline | | | }*** | | | ------------- ----------------- | | | ------ | | | ***t*** | | | | | | -- | | | -- | | | | | | ***Speed* **is the rate of change | | | of *distance* with time. | | | | | | **Average speed** is the rate of | | | change of distance with time. | | | | | | **Average velocity** is the rate | | | of change of displacement with | | | time. | | | | | | In calculus, **Instantaneous | | | speed** is the first derivative | | | of distance with respect to time