Preparation for Physics I-1 REVIEW OF EXPONENTS

Transcription

1 Preparatio for Physics I-1 REVIEW OF EXPONENTS Defiitio: If a is a real uber ad is a atural uber, a a1 44 a a L factors of a where a is the base ad is the expoet. Defiitio: For ay atural uber ad ay ozero real uber a, the egative expoet is defied by - 1 a. a RULES OF EXPONENTS Product Rule: If a is ay real uber ad ad are atural ubers, the + a a a. Quotiet Rule: If a is ay ozero real uber ad ad are ozero itegers, the a - a. a Zero Expoet: If a is ay ozero real uber, the a 0 1. Note: 0 0 is udefied. Power Rules: If a ad b are real ubers ad ad are itegers, the ( ) a a ( a b) a b a b a b (b 0). ( ) Note: The power rule, a b a b, applies whe raisig a uber with uits to a power. The volue of a cubical box that is ft o each side is ( ft) ft 8 ft. COMMONLY CONFUSED EXPRESSIONS (1) -a ad (-a) do ot ea the sae thig. () ab is ot the sae as (ab). () (a + b) is ot the sae as a + b.

2 Preparatio for Physics I- VERIFYING THE RULES OF EXPONENTS The defiitios of expoets ad egative expoets ay be used to verify the rules of expoets. Exaple: The product rule. a a a a a a a a 144 L 144 L factors of a factors of a a1 44 a a L a + factors of a + Exercises A ad B will be doe i class. They should be icluded with the rest of the exercises fro this worksheet. EXERCISE A. Verify the quotiet rule, the zero expoet rule, ad the power rules usig the defiitios. CHECKING THE COMMONLY CONFUSED EXPRESSIONS If two expressios ca be show by ay oe exaple to yield differet results, the the two expressios do ot ea the sae thig. Exaple: (1) -a ad (-a) do ot ea the sae thig. We ca illustrate this by choosig a ad. Now, -a , while (-a) (-) (-). (-) 9 Sice this exaple shows that the two expressios yield differet results, the two expressios do ot ea the sae thig. Note: You ay be able to fid exaples where the two expressios yield the sae result, eve if the two expressios do ot ea the sae thig. You oly eed to fid oe exaple where the expressios yield differet results to show that the expressios do ot ea the sae thig. If a ad, the -a ad (-a) (-) (-). (). (-) -7 Although i this exaple both expressios yield the sae result, the couterexaple give above (a ad ) shows that the two expressios do ot ea the sae thig. EXERCISE B. Fid exaples (uerical values for a, b ad ) that illustrate cooly cofused expressios () ad ().

3 Preparatio for Physics I- EXERCISE C. USING THE RULES OF EXPONENTS. Siplify each expressio. Assue that all variables used as expoets represet itegers ad that all other variables represet ozero real ubers. Write all aswers with oly positive expoets. Evaluate wheever possible. 1. a. (-1) 1 b. (-1) c. (-1) d. (-1) 4 e. (-1) 5 f. (-1) 6 g. Do you see a patter i the sigs here? Explai this result i words. Does this patter oly work for -1 or does it work for ay egative uber? h. (-1) if is odd i. (-1) if is eve j. (-) if is k. (-) - if is. a. x x 5 x b. ( x 4 ) 4x ( ) c. ( 5a 6 b 4 )( a b 5 ) ( ) 4 z d. 5( r )( r 5 ). a. b. p c. 4. a a. ( ) 4 b. y y 8 ([ x x] ) + x 0 c. t 6 t 4 + b. ( f ) 4 ( f 5 ) (9 f ) 6 6. a. ( ω ) (ω ) 4 (ω 5 ) 6 b. λ6 + λ + λ 4 7. a. (p + q) 4 ( p + q) b. 4 ( 5 ) ( p + q) 6 (4 ) (6 7 ) λ (4b)1 d. (4b) ( c d ) ( 8. a ( 6 c d ) cd 4 ) ( 6 ) ( 7 8v w 4 v w ) b. 5 4 ( v w ) 4 5

4 Preparatio for Physics I-4 REVIEW OF ROOTS AND RADICALS Defiitio: For ay real uber x ad atural uber, the real uber r is a th root of x if r x. x deotes the th root of x. x is the radicad. is the radical sig or the radical. is the idex, or order, of the radical. Square roots: If x is a real uber ad x > 0, x is the positive (priicipal) square root of x. x is the egative square root of x. If x is ay real uber, the x x. Fractioal expoets: The th root of x ca be deoted by x 1/ or x. Therefore, x 1/ For all positive itegers, 0 1/ 0. x. If x is positive ad is a eve positive iteger, the x 1/ is positive. If x is positive ad is a odd positive iteger, the x 1/ is positive. If x is egative ad is a eve positive iteger, the x 1/ is ot a real uber. If x is egative ad is a odd positive iteger, the x 1/ is egative. RULES OF RADICALS Product Rule: If x ad y are ay positive real ubers ad is a atural uber, the x. y xy or x 1/. y 1/ (xy) 1/. Note: This rule is frequetly used whe workig with quatities with uits. Suppose you wish to kow the legth of a edge of a square with a area of 5 i. The legth is related to the area by, L A, so L A 5i 5 i 5 i. Quotiet Rule: If x ad y are ay positive real ubers (y 0) ad is a atural uber, the x x y y or x y 1 x y 1/ 1/.

5 Preparatio for Physics I-5 Power Rule: If ad are positive itegers, the x / 1/ 1/ ( x ) ( x ) provided all idicated roots exist. If x / exists ad x 0, the / 1 x / x. Sice radicals ca be writte i fractioal expoet for, these ad other rules follow fro the defiitios ad rules of expoets. COMMONLY CONFUSED EXPRESSIONS EXERCISES 1/ 1 (1) x x () x + y x + y 1. Fid exaples that illustrate cooly cofused expressios (1) ad ().. Without usig a calculator, fid the uerical equivalet of (that is, solve) each of the followig. a b. 64 1/ - 6 1/ c d. 15 1/ /4 e. 8 9 f. 15 1/ x 5 1/ g h. (10 7 ) 1/ x (10 10 ) /5. Siplify each expressio if possible. Assue that all idicated roots exist. a. a + b b. σ c. σ σ d. 4 σ σ σ e. 6 ( ) 6 7x y ( 8x y ) f. ( x y) g. (α + β ) 6 ( 6 )( ( )) 6x 4 y ( ) 6 ( λ + μ) (α + β) 6 h. ( λ + μ) 9 ( λ + μ)

6 Preparatio for Physics I-6 i. (λ + μ) 6 + (λ + μ) (λ + μ) 9 j. ( + T ) ( )( ) 5 4 ( a + b) ( a + b) k.. ( a + b) ( p q) ( (p q ) ) 5 ( p q ) 6 [ ] [ ] 1 z ( z + T ) ( )( ) ( ab) ( ab) l.. 4 ( ab) ( 6x y ) ( (x y ) 4 ) (1x y ) 4 5

7 Preparatio for Physics I-7 REVIEW OF EXPONENTS AND ROOTS More Practice Exercises Siplify each expressio. Assue that all variables used as expoets represet itegers ad that all other variables represet ozero real ubers. Write all aswers with oly positive expoets. Evaluate wheever possible. 1. ( β α ) 4 ( β 6 α ). ( v 6 q ) 4 ( v 6 q ) 4. ( - )( - ) 4. (-c b 4 ) ( k p ) k p ( ) 1 ( ) 1 ( f ) ( f ) f ( ab ) (a b ) (6 f ) (a 5 b) w7 + w 5 w 4 w (σ + δ )5 (σ + δ ) 7 (σ + δ) 8 (σ + δ) d4 ( d 6 g ) 4 (4d ) 10g ( ) x y x x y ( 7 ) ( r )( r ) r a 16 b 1 4 ab ( ) ( ) 18. x y xy xy 19. k 1 k k ( s ) 6 r r t s t

8 Preparatio for Physics I-8 Algebraic Equatios Types of Equatios ad Solutios Solutios to Equatios are values for the variable or variables that ake the equatio true. Exaples: x is a solutio to the equatio x + 5, because + 5 x ad y is a solutio to the equatio y x 1, because 1 x is a solutio to the equatio x 9, because 9 x - is also a solutio to the equatio x 9, because (-) 9 Note: Whe we list ultiple solutios to a sigle equatio, we separate the solutios with the word or. Sice x 9 is true if x or x -, we say that the equatio x 9 is solved if x or x -. I this case we ca save writig with the plus-or-ius-sybol, ±, ad write the solutios to x 9 as x ±. Not all equatios have solutios. I this class, we will restrict ourselves to real solutios to equatios. For exaple, x -1 has o real solutios. A equatio with o solutios is called a cotradictio. A exaple of a cotradictio is the equatio x x +. There is o value you ca substitute for x ad have both sides of the equatio be the sae value. Soe equatios are true for soe, but ot all, values of the variable or variables. Such a equatio is called a coditioal. A exaple of a coditioal is x x + y. This equatio is true provided y 0, but is ot true if y is ay other value. Other equatios are true for all values of the variable or variables. Such a equatio is called a idetity. With a idetity, oe side of the equatio ca be rearraged ito the other side of the equatio. Exaples iclude distributio, (x - ) x - 6, obtaiig a coo deoiator, x 1 x+ 1 +, defiitios, a a a a, ad rules, a a a A. Classify each of the followig equatios as a idetity, a coditioal or a cotradictio. Assue all variables are positive real ubers. 1. A B + C AD + CB D BD. A B A + B +. C D C + D λ α + β λ α + λ β 4. α + β λ α λ + β λ 5. (G+Q) (k +d ) (k+d ) (G+ Q) 1 6. σ σ φ θ φ θ 7. σ σ θ θ φ φ 8. R σ S + T β RS + RT σβ 9. R σ S + T β Rβ σs + T C C C 1 C C 1 + C

9 Preparatio for Physics I-9 Solvig a Equatio for a Ukow Variable Two equatios with the sae solutios are said to be equivalet. A exaple of a pair of equivalet equatios is x + 5 ad x. Both of these equatios are true oly if x. Give ay equatio (x ), you ca obtai a equivalet equatio by doig ay of the followig: Add the sae thig to both sides of the equatio, x + + or x + 5 Subtract the sae thig fro both sides of the equatio, x or x - 1 Multiply both sides of the equatio by the sae thig (except 0!), x or x 6 Divide both sides of the equatio by the sae thig (except 0!), xπ π or 1 x 1 Of course subtractio is the sae as additio of a egative ad divisio is the sae as ultiplicatio by a reciprocal, so we do t really eed all of the exaples, as log as we reeber ot to ultiply or divide by zero. The equatios x, x + 5, x - 1, x 6, ad 1 x 1 are all equivalet, because they are true oly for x. That is to say, x is the oly solutio to each of these equatios. If we are asked to solve the equatio x + 5 for x, we would subtract fro both sides ad have x which siplifies to x. To solve x 6 for x, we could ultiply both sides by ½. (½)(x) (½)(6) which siplifies to x. Istead of ultiplyig by ½, we could have divided by ad obtaied the sae result. Soeties to solve, we ust perfor ore tha oe step. Suppose we wish to solve the equatio a + b 1 for a. This equatio cotais two variables, a ad b. Whe we solve for a, we are cosiderig b to be a kow quatity. We ay have a sigle value for b or we ay have a list of values, i.e. Deterie the value of a whe b is 1,,, 4, ad 5. (We will see i a later sectio that this ca be useful whe graphig the solutios for a equatio with two variables, where we will solve a equatio for oe variable ad plug i values for the other variable to deterie ordered pairs to plot.) a + b 1 a + b b 1 b a 1 b 1 1 a ( 1 b) b a 6 Not all solutios ca be foud with these siple steps. Soeties we ust use expoets or roots, the quadratic forula, or other ethods to fid a solutio. Exaple Requirig a Root: The solutios to the equatio x 9 are foud by takig the square root of each side of the equatio. Whe we use a square root, we have to reeber that we will have both a positive ad a egative solutio, x ± i this case. Exaple Requirig a Expoet: The solutio to the equatio b 9 is foud by squarig both sides of the equatio. The result i this case is b 81.

10 Preparatio for Physics I-10 Quadratic forula: The quadratic forula is frequetly used to solve algebraic probles i physics. b ± b 4ac Whe the quadratic forula is writte as x, it is the solutio to the quadratic a equatio ax + bx + c 0 for the variable x. The equatio ax + bx + c 0 cotais 4 variables, but oly x is cosidered ukow. Siple Exaples: Solve x 5x + 0 for x. ( 5) ± ( 5) 4 5 ± 9 5 ± 8 1 x or or Solve 4h 1h 0 for h. ( 1) ± ( 1) ± h or or Notice that i the secod exaple, c was zero i the quadratic forula because there was o costat ter i the quadratic equatio 4h 1h 0. Whe usig the quadratic forula, b ad/or c ca be zero but a caot. Why? The coefficiets a, b ad c are ot always give i the origial equatio ad ca cotai variables. More Coplex Exaple requirig the Quadratic Forula: A right triagle has a base b, a height h ad a hypoteuse that is 5 ties the differece betwee the height ad base. Deterie the height of the triagle. We will lear i a later sectio the steps for settig up geoetric word probles. I this case we will draw a picture ad apply the Pythagorea Theore. h 5 h b b b b + h + h ( 5h b ) 5h 0 4h 50hb + 5b 50bh + 4b We ow apply the quadratic forula with 4 as a, (-50b) as b ad 4b as c. h ( 50b) ± ( 50b) 5b ± b 4 So the height is b 4 b b 50b ± ( 5 4 ) 5b ± b ( 5 4)( 5 + 4) b b b 4 ( 5 ± ()( 1 49) ) ( 5 ± 7) ( ) or ( 18) b or b or 4 b b b 4

11 Preparatio for Physics I-11 B. Solve for the specified ukow variable i each of the followig exaples: 1. Cosider F to be ukow: A + B F G + t. Cosider t to be ukow: A + B F G + t. Cosider a to be ukow: θ + r a 4. Cosider λ to be ukow: α + β λ 5. Cosider α to be ukow: α + β 6. Cosider d to be ukow: a 0 λ α + β α β λ λ b 4cH c c + d 7. More Challegig: Redo #6, cosiderig c to be ukow. Actual Physics Exaples: Here are soe actual physics equatios that you will likely ru ito. Notice that you ca do the algebra without kowig what the equatios ea! (You will eed to uderstad the physics to set up the equatios so that they are ready to solve.) Notice that, i soe cases, you do ot have ubers to "plug i," while i others you do. I the latter case, however, it is usually advisable to rearrage the sybols before puttig i the uerical values. This serves at least two purposes: 1. It akes it easier to fid istakes. If you put i the ubers early o, erroeous algebraic steps are ot easy to fid.. You will ed up with a geeral solutio. A geeral solutio ca be used to ivestigate questios like "What if the object started with twice the speed?, "What agle gives the axiu distace?, "What if there were o frictio?, ad so o. A geeral solutio is also useful i lab situatios. Be ready to use the two algebraic quick steps, to group ters ad factor, to raise both sides to a power or take roots of both sides, ad to use the quadratic equatio.

12 Preparatio for Physics I-1 C. Solve for the specified ukow variable i the followig physics equatios. Assue that all other variables are kow. If uerical values are give put the i at the ed to obtai a uerical result. [Note: I these cases the kow variables have a uerical value ad uits. Siplify the uits as you would variables usig the rules of expoets ad roots.] 1. t?: v 1 0 /s,v 10 /s, a 10 /s : v v 1 + at. T 1?: Q c(t T 1 ). r? : qvb v r 4. L?, g 9.8 /s, T 1.0 s : T π L g 5. R eq?, R 1 Ω, R 4 Ω: 1 R R 1 R eq 6.?, F 0.0 N, T 4.0 N, µ k 0.15, g 9.8 /s, a.0 /s : F μ k g T a 7. r? : 1 v G M r 1 kx 8. t?, y 5.0, v 1 0 /s, g 10 /s : Δy v 1 t 1 gt If you ca work through these fluetly (accurately ad quickly), you are well o your way towards atheatical preparedess for physics. If ot, you ca get there. Just practice, practice, practice util you get the gae.

13 Preparatio for Physics I-1 Preparatio for Proble Solvig I Before begiig proble solvig (word probles) it is ecessary to develop soe basic skills to be applied to the aalysis of probles. First of all, oe ust be able to read the stateet of a proble ad traslate it ito sybols ad pictures. These exercises are desiged to develop these skills. Words to Pictures Sketch a diagra to represet each situatio described. Exaple: A rectagular block is placed with its sallest side o top of a cube. (1) A ladder is leaed agaist a wall so that its top is twice as far fro the floor as its botto is fro the wall. () A chair is tipped back o its rear legs so that they ake a agle ø with respect to the floor. () A rectagle has a legth that is twice its width. (4) A sall ball is restig o top of a large box that is sittig o a table. (5) Three blocks are stacked o a iclied plae with the sallest block i the iddle ad the largest block o top. (6) Two trai statios are located 00 iles apart. Oe trai leaves statio A headig toward statio B at the sae tie a secod trai leaves statio B headig toward statio A. The two trais pass each other 50 iles fro statio A. (7) A 10-kg block ad a 5-kg block are joied by a strig that passes over a pulley hug fro the ceilig. The 5-kg block is hagig lower tha the 10-kg block. (8) A flat board is sittig o top of a rectagular box. A secod box is restig o top of the board.

14 Preparatio for Physics I-14 Words to Algebra Assig a sybol to represet each quatity. Covert the setece ito a algebraic expressio (equatio). Exaple: the desity of a object is defied to be its ass divided by its volue. Let ρ desity, ass, ad V volue, the ρ /V. (1) The teperature expressed i kelvi is 7.15 greater tha the teperature expressed i degrees celsius. () The average of three distaces is the su of the distaces divided by three. () The distace traveled by a car ovig at a costat speed is the product of its speed ad the tie of travel. (4) The area of a triagle is oe-half the product of its base ad its height. (5) Object A is five ties as log as object B. (6) If Mary were 0 pouds heavier she would be twice as heavy as Eric. (7) A object starts fro rest ad travels a certai distace i a straight lie with a costat acceleratio. The tie this takes is the square root of the quatity twice the distace divided by the acceleratio. (8) The frictio force actig o a block slidig dow a iclie is oe-teth of the weight of the block. (9) If the legth of block 1 were tripled it would be equal to half the legth of block. (10) The legth of a rectagle is eight iches ore tha triple its width.

15 Preparatio for Physics I-15 Preparatio for Proble Solvig II Aother iportat skill i doig physics probles is to be able to visualize the physical process beig aalyzed. It is difficult to costruct diagras or algebraic relatios if you caot visualize what s goig o. These siple set-ups will provide a foudatio for visualizig both echaical systes as well as odels for o-echaical systes. Words to Set-Ups Use the set of objects provided by your istructor to physically set-up what is described here. Work i groups of at least two ad ake sure everyoe i you group agrees with the fial set-up. Have your istructor check that your set-up precisely atches the descriptio. A C B (1) A large rectagular block is placed o the table with its sallest face i cotact with the table. A idetical block is placed o top of the first block with the secod block s largest face o top of the first block. [How ay correct variatios of this set-up ca you coe up with?] () Three blocks are stacked o a iclied plae with the sallest block i the iddle ad the largest block o top. () Place a type C-block so that oe of its largest faces is i cotact with the table. Place a A-block o the table with its sallest face o the table. Place the two blocks such that the sallest face of the C block is two legths (logest diesio) of the large block away fro the secod largest face of the large block. (4) A B-block is placed with its largest face o a iclied surface. A C-block is placed o top of the B-block with its sallest face touchig the B-block. The rap is the iclied to the axiu agle that ca be achieved without the C-block topplig. (5) A C-block is placed o top of a B-block so that it lies across the diagoal of the largest face of the B-block. The C-block is cetered o this diagoal. A A-block is placed sall-face-dow o top of the C-block ad is rotated so as to ake a right agle with the C-block. [How ay right agles does the C-block ake with the A- block?] (6) A A-block is o the table with its largest side dow. Place a B-block o its largest side so that its sallest side is touchig the sallest side of the A-block.

16 Preparatio for Physics I-16 (7) Place C-blocks o the iclied plae, with the sallest faces touchig the iclied plae. The two C-blocks should be separated by a distace equal to the logest side of a B-block. Place a A-block o top of the C-blocks with the largest face of the A-block touchig the tops of the C-blocks. The ext two exercises have you costruct odels that show key states of physical processes. (8) Two trais are located 00 iles apart. Oe trai (use a sigle A-block) leaves statio A headig toward statio B at the sae tie a secod trai (use a B-block) leaves statio B headig toward statio A. The two trais pass each other 50 iles fro statio A. Usig two A-blocks ad two B-blocks, costruct a physical odel that shows both states (istats) described. (9) A Califoria codor flyig over the desert adires the beautiful hues ad shadig of the soft dusk light. She looks dow ad sees telephoe poles (C-blocks) o a straight, horizotal road. Fro experiece, the codor kows that the distace betwee the poles is equal to the height of oe pole. She spots two cars (use B-blocks) approachig each other with costat speeds. Oe car is at pole #1, the other is at pole #7. She otices that the car at pole #1 is travelig twice as fast as the car at pole #7, so without a calculator! she quickly predicts where the cars will pass. Place two B-blocks i the observed positios. Leave the there ad place two ore B-blocks i the predicted passig positios. Now it s your tur! (10) Write (i words) the descriptio of three differet challege set-ups for your parter(s). Oe descriptio should be of ustacked blocks o a horizotal surface describe distaces ad orietatios. The secod descriptio should be of blocks stacked o a horizotal surface. The third descriptio should be of blocks stacked o a iclie. Exchage writte descriptios with your parter(s). Costruct the set-ups that your parter has described (o paper) for you.

17 Preparatio for Physics I-17 Review of Expoetial ad Logarithic Fuctios Defiitio: For b > 0 ad b 1, ad all real ubers x, y b x is a expoetial fuctio with "base b." Property: For b > 0 ad b 1, if b x b y, the x y. Defiitio: For b > 0 ad b 1, ad all positive ubers x, y log b x eas the sae as x b y. Properties: If b > 0 ad b 1, the log b b x x. Rules of Logariths If b > 0 ad b 1 ad x > 0, the. b log b x For b > 0 ad b 1, log b b 1 ad log b 1 0. Product Rule: If x, y, ad b are positive ubers, b 1, the log b xy log b x + log b y. Quotiet Rule: If x, y, ad b are positive ubers, b 1, the log b (x/y) log b x - log b y. x Power Rule: If x ad b are positive real ubers, b 1, ad if is ay real uber, the log b x log b x. Verifyig the rules of logariths. To prove the product rule, rewrite the logarithic stateets i expoetial for log b x eas b x ad log b y eas b y. So, by substitutio, ad usig the product rule for expoets, xy b. b b +. Rewrite this i logarithic for, log b xy +. Now substitute for ad to get log b xy log b x + log b y. These rules ad properties ca be used to solve algebra probles as show i the followig exaple. Exaple: Solve the followig equatio for x: 5 x + 5. Take log 5 of both sides of the equatio: log 5 5 x + log 5 5 This leads to x + log 5 5 which becoes x +. Solvig gives x -1.

18 Preparatio for Physics I-18 Cooly Cofused Expressios (1) log b (x ± y) is ot the sae as log b x ± log b y. () log b (xy) is ot the sae as (log b x) (log b y.) EXERCISES 1. a. For each stateet give i expoetial for, rewrite it i logarithic for. i. x 4 ii ,000 iii. 4-1/64 b. For each stateet give i logarithic for, rewrite it i expoetial for. i. log 8 ii. log 10 1,000,000 6 iii. log Solve the followig equatios for the give variable. a. x 4 b. 9 x 7 c. 4 x 64 d. x 1 9 e. x 1 8 f. 6 x 1 6 g. x+ 16 h. 16 (x+1) / Probles ad 4 ay be doe as i-class exercises. Check with your istructor.. Show why for the first property (For b > 0 ad b 1, if b x b y, the x y.), the stateet "b 1" is ecessary. 4. Verify the quotiet rule ad the power rule. 5. Evaluate the followig. a. log 10 1,000 b. log c. log 10 1 d. log 8 64 e. log 5 15 f. log (1/7) g. log 5 5 h. log 7 1/ 6. Solve each equatio for the give variable. a. y log b. log 5 x - c. log x 9 1/ d. log c 15 - e. log 4 x 5/ f. log µ 0 g. log p h. log 1/ f 1 7. Write each of the followig as a sigle logarith. Assue all variables represet positive real ubers. Siplify as uch as possible. a. log 6 + log 6 b. log log 10 5 c. log 5 + log Fid exaples that illustrate the cooly cofused expressios.

19 Preparatio for Physics I-19 Applicatios Physics requires the applicatio of ath skills to solve probles. I these exercises, we will apply the ath cocepts reviewed i this sectio to soe probles. Geoetric Applicatios Soe probles ecoutered i the real world require the kowledge of siple geoetric forulas. Exaple 1 Suppose you are asked to deterie the width of a rectagle which has a legth which is twice its width ad has a perieter of 18 feet. To solve this we will follow a set of proble solvig steps. 1. Draw a picture This proble is about a rectagle, so we draw a rectagle:. Assig a sybol to each kow ad ukow quatity ad label your diagra with those Give: P perieter of rectagle 18 ft Ukow: w width of rectagle ad L w legth of rectagle w w. Deterie the relatioship betwee the kow ad ukow quatities P L + w *w+*w or P w + w + w + w 4. Solve sybolically P 6w w P/6 5. Substitute kow values ad evaluate uerically w P/6 (18 ft)/6 ft This very siple exaple illustrates our proble solvig steps. CSM Physics Departet 005

20 Preparatio for Physics I-0 Exaple Deterie the area of a equilateral triagle with sides of legth L. 1. Draw a picture. Assig a sybol to each kow ad ukow quatity ad label your diagra with those Give: L legth of each side; b base of triagle L Ukow: h height of triagle Ukow: A area of triagle L L h L. Deterie the relatioship betwee the kow ad ukow quatities Cosiderig the right triagle o the right with hypoteuse L ad legs h ad ½ L: h + (½ L) L. A ½ b h ½ L h 4. Solve sybolically Fid h: h + ¼ L L h ¾ L Copute A: A 1 L h 1 L L L 4 h L 4 L 5. Substitute kow values ad evaluate uerically I this case there is o uerical value to substitute for L. L A is the fial result. 4 CSM Physics Departet 005

21 Preparatio for Physics I-1 Expoetial Growth ad Decay Applicatios I ay areas of sciece as well as other fields, the growth rate of a quatity is proportioal to its curret value. For exaple, the growth rate of a populatio of buffalo is proportioal to the uber of buffalo preset i the absece of ay predators. I such a case, there is a expoetial growth. Exaple Suppose the populatio of a herd of buffalo i Motaa doubles every years. Today, there are 0 buffalo i the herd. How ay years will it take before there are 640 buffalo i the herd? There is o picture to draw i solvig this proble. So, we will begi our process by assigig sybols to the ukow ad kow quatities. Our proble solvig steps are as follows: 1. Assig a sybol to each kow ad ukow quatity Ukow: t tie for populatio to reach 640 Give: t d tie for populatio to double yr Give: N 0 uber of buffalo preset today 0 Give: N uber of buffalo preset at tie t 640. Deterie the relatio betwee the kow ad ukow quatities t t td t d 0 N0 N N. Solve sybolically N N 0 t t d t t d N N 0 t t d log N N 0 t t d log N N 0 4. Substitute kow values ad evaluate uerically N t log N ( yr) log ( yr) log ( ) ( yr)( 5) 15yr t d May processes i sciece also ivolve expoetial decay. May cheical reactios have rates that are proportioal to the aout of reactat preset. For exaple, the decopositio of caffeie i the body occurs with a half-life of 6 hours. This eas if you have 00 g of caffeie preset i your body at 8 a, you will have 150 g preset 6 hours later at p ad 75 g preset after aother 6 hours at 8 p ad so o. CSM Physics Departet 005

22 Preparatio for Physics I- Exaple 4 Suppose you take a No-Doze pill which cotais 00 g of caffeie. How log will it take for the aout of caffeie i your body to drop to 5 g? 1. Assig a sybol to each kow ad ukow quatity Ukow: t tie for aout of caffeie to reach 5 g Give: T 1/ half-life of caffeie 6 h Give: 0 begiig aout (ass) of caffeie 00 g Give: aout (ass) of caffeie preset at tie t 5 g. Deterie the relatio betwee the kow ad ukow quatities 1 0 t T 1/. Solve sybolically 1 0 t T t T 1/ 1/ 0 t T 1/ log 0 t T 1/ log 0 4. Substitute kow values ad evaluate uerically 5μg 1 t T log 0 00μg 8 ( 6 h) log ( 6 h) log ( 6 h)( ) 18h 1/ Exercises 1. Deterie the area of a square, which has a diagoal with legth.. A rectagle has a legth which is feet less tha twice its width. The area of the rectagle is 14 ft (14 square feet). What is the legth of the rectagle?. The populatio of rabbits i a couty is foud to triple every 4 years. If there are curretly 100 rabbits i the couty ad this tred cotiues, how ay rabbits will there be i 16 years? 4. Epiephrie (cooly called adrealie) is a horoe released fro the adreal glads i eergecy situatios or whe dager is sesed. Epiephrie has a half life of iutes. If your body releases 400 icrogras of epiephrie whe a deer jups i frot of the car you are ridig i, how log will it take for the aout of epiephrie i your body to drop to 5 icrogras? CSM Physics Departet 005

23 Preparatio for Physics I- Review #1: Basic Math All exercises should be doe without calculators. A. Siplify. Assue all variables represet o-zero real ubers. Write all aswers with oly positive expoets. 1. ( - ). ( ). (/) f x f x 9. a a 4 a 6 a ( b)) (b ) ω ρ ( 6) ω 8 ρ 5 1. (si φ) (siφ ) 8 (siφ ) 6 1. (x + y) (x + y) (x + y) 8 (x + y) 5 (x + y) ( a b c 5 ) 4 a b 4 c ( ) ( ac ) ( a 6 b ) 4 ( b c 5 ) 15. [(ψ + λ ) (ψ + λ ) (ψ + λ) 15 + (ψ + λ ) 8 (ψ + λ) ] 0 B. Siplify. Assue all variables represet positive real ubers ( x 10 )9 x ( ) (x + y) 5 (x + y) ( 100 )10 ( 6 )1000 ( ) 11. r r 1 4r 4 z 1 x 5 1. z 1 x 5 1. σμ σ 5 μ σ 8 μ 4 σ 7 μ C. Rewrite each relatio usig logarithic otatio. 1. x b p E 4. 4 x 64

24 Preparatio for Physics I-4 D. Rewrite each relatio i expoetial for. 1. log b E. log log x y 4. log 8 (1/8) -1 E. Write each of the followig as a sigle logarith. Assue all variables represet positive real ubers. Siplify as uch as possible. 1. log b μ +log b 7. log 4 +log log x - log x q 4. xlog b - log b x F. Solve each equatio for the idicated variable. Siplify your results Solve for μ : μ + Δ Δ + μ. Solve for q: f p q. Solve for t : x + at 1 vt 4. Solve for β : β φ β + 4φ Solve for v : gh v + gy 6. Solve for a: Mg - Ma μg + a. 7. i. Solve for t: x f x i + v i t + ½ a t ii. Fid t whe x i 0, x f 18.75, v i 0 /s, ad a 1.5 /s. iii. Fid t whe x i 18.75, x f.5, v i 7.5 /s, ad a -.0 /s.

25 Preparatio for Physics I-5 Buildig Physical Ituitio - Pushig Blocks This exercise is desiged for you to develop your ability to record ad couicate observatios you ake i a effective aer. Studets should work i pairs. Equipet: Each pair of studets will eed a woode block ad a writig istruet with a eraser for pushig the block. Each group eber should record all observatios i ik o separate otebook paper. You will had i this paper at the ed of the activity. 1. Place the block o the table with its largest face dow. Place the eraser of your pecil at the positio labeled "a" with the pecil parallel to the table ad getly push. Observe the otio of the block. Record your observatio usig a coplete setece. Repeat for each positio ( b through f ) o the block.. Now place the block o the table as show i figure. For each poit, g, h ad i, place the eraser of your pecil at the positio with the pecil parallel to the table ad getly push. Observe the otio of the block. Record your observatio usig a coplete setece. Repeat the process with a harder push. Does the otio of the block deped o how getly (slowly) or how hard (quickly) you push it? Describe ay differeces.

26 Preparatio for Physics I-6 Buildig Physical Ituitio Stacked Blocks The goals of this activity are to cotiue to develop your observatio skills ad to begi to becoe aware of forces (pushes ad pulls) ad their effects o objects. You will ake predictios, test your predictios, ad explai differeces betwee your predictios ad your observatios. Use your results to ake predictios about ew situatios. Studets should work i pairs. Equipet: Each pair of studets will eed five woode blocks ad a writig istruet with a eraser for pushig a block. Each group eber should record all predictios ad observatios i ik. Part A. Stack two blocks o the table as show. Call the botto block block 1 ad the top block block. block block 1 1. Predict what will happe whe you push softly/slowly o the ed of block. Push softly/slowly o the ed of block. Record your observatios below. Did your observatios agree with your predictios? If they did ot agree, ca you offer a explaatio? CSM Physics Departet

27 Preparatio for Physics I-7. Predict what will happe whe you push softly/slowly o the ed of block 1. Push softly/slowly o the ed of block 1. Record your observatios below. Did your observatios agree with your predictios? If they did ot agree, ca you offer a explaatio?. Predict what will happe whe you push hard/fast o the ed of block 1. Push hard/fast o the ed of block 1. Record your observatios below. Did your observatios agree with your predictios? If they did ot agree, ca you offer a explaatio? CSM Physics Departet

28 Preparatio for Physics I-8 Did block ove forward or backward relative to block 1? Did block ove forward or backward relative to the table? 4. Discuss the differeces i the results of 1,, ad. Explai why these differeces occurred. Part B. Stack a third block (block ) o top of block. 1. Predict what will happe whe you push softly/slowly o block. Push softly/slowly o the ed of block. Record your observatios below. Did your observatios agree with your predictios? If they did ot agree, ca you offer a explaatio? CSM Physics Departet

29 Preparatio for Physics I-9. Predict what will happe whe you push hard/fast o block. Push hard/fast o the ed of block. Record your observatios below. Did your observatios agree with your predictios? If they did ot agree, ca you offer a explaatio? Part C. Stack two additioal blocks o top of blocks 1,, ad. Feel the differece i the forces required to slide differet blocks. For exaple, how uch harder or easier is it to slide block tha to slide block 5 (top block)? Hypothesize what factors are resposible for your results. Record your observatios, hypotheses, ad ay additioal coets. Part D. If tie perits, develop soe additioal experiets. Make predictios ad test the. Record your observatios, hypotheses, ad ay additioal coets o a separate paper. CSM Physics Departet