Motion In One Dimension. Graphing Constant Speed
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1 Moion In One Dimenion PLATO AND ARISTOTLE GALILEO GALILEI LEANING TOWER OF PISA Graphing Conan Speed Diance v. Time for Toy Car (0-5 ec.) be-fi line (from TI calculaor) d = Diance (cm) Conan peed i he lope of he (be fi) line for a diance v. graph. 600 peed = diance 400 = 208 cm (3 ig fig) Remember, he andard meric uni for lengh i he meer! 200 = cm 10 2 m m = cm 5.0 Time () 1
2 Graphing Average and Inananeou Speed Diance v. Time for Toy Car (0-0.5 ec.) Average peed i he lope of a ecan line for a diance v. graph. average peed = diance mm avg = avg = 1190 mm 10 3 m 1 mm = 1.19 m Diance (mm) click for apple (0.5, 350) (0.13, 0) Time () Inananeou peed i he lope of a angen line for a diance v. graph. in. peed = diance (a 0) = mm = 946 mm approximae lope a = m 1 mm = m be-fi quadraic (from TI calculaor) d = ange line lope a = 0.25 (from calc.) d = Diance, Poiion and Diplacemen Diance (d) The lengh of a pah raveled by an objec. I i never negaive, even if an objec revere i direcion. Poiion (x or y) The locaion of an objec relaive o an origin. I can be eiher poiive or negaive Diplacemen ( x or y) The change in poiion Δx = x f x i of an objec. Alo can be poiive or negaive. Δy = y f y i One dimenional moion A B C x(m) Wha i he diance raveled if an objec ar a poin C, move o A, hen o B? d = d 1 + d 2 = = 12 m 2. Wha i he diplacemen of an objec ha ar a poin C and move o poin B? x f x i = -2 (+4) = -6 m 3. Wha i he diplacemen of an objec ha ar a poin A, hen move o poin C and hen move o poin B? x f x i = -2 (-5) = +3 m Two dimenional moion 4. Wha i he diance raveled and he diplacemen of he peron ha ar a poin A, hen move o poin B, and end a poin C? d = = 7 m; x = = 5 m 2
3 Diance and Poiion Graph d (m) Diance v. Time x (m) Poiion v. Time poiive CAR B: conan poiive velociy negaive CAR C: conan negaive velociy Diance graph how how far an objec ravel. Speed i deermined from he lope of he graph, which canno be negaive. Poiion graph how iniial poiion, diplacemen, velociy (magniude and direcion). Tha why poiion graph are beer! Remember, all of hee graph how conan peed. (How do you know?) Average Speed v. Average Velociy Average peed i he diance raveled divided by elaped. average peed = diance raveled elaped avg = d Average velociy i diplacemen divided by elaped. average velociy = diplacemen inerval v avg = Δx Example: A priner run 100 m in 10, jog 50 m furher in 10, and hen walk back o he finih line in 20 econd. Wha i he priner average peed and average velociy for he enire? avg = d = 200 m 40 = 5 m v avg = Δx = 100 m 40 = 2.5 m d (m) lope = ave. peed x (m) lope = ave. velociy
4 Inananeou Speed and Velociy Inananeou peed i he how fa an objec move a an exac momen in. Inananeou velociy ha peed and direcion. inananeou peed = diance a approache zero Honor: d = lim 0 inananeou velociy = diplacemen a approache zero Δx v = lim = dx 0 d Inananeou peed (or velociy) i found graphically from he lope of a angen line a any poin on a diance (or poiion) v. graph. d (m) lope of angen = inananeou peed x (m) lope of angen = inananeou velociy ign of lope = ign of velociy The Phyic of Acceleraion Acceleraion i how quickly how fa change how fa mean velociy how fa change mean change in velociy how quickly mean how much elape Acceleraion i defined a he rae a which an objec velociy change. acceleraion = change in velociy a avg = Δv Acceleraion ha uni of meer per econd per econd, or m//, or m/ 2. Acceleraion i conidered a a rae of a rae. Why? Meric (SI) uni m or m 2 4
5 Type of Acceleraion v (m/) Velociy v. Time v (m/) Velociy v. Time Conan Acceleraion Conan acceleraion i he lope of a velociy v. graph. (Sound familiar?! Compare o, bu DO NOT confue wih conan velociy on a poiion v. graph.) Varying Acceleraion Average acceleraion i he lope of a ecan line for a velociy v. graph. Inananeou acceleraion i he lope of a angen line for a velociy v. graph. (Again, compare o, bu DO NOT confue wih average and inananeou velociy on a poiion v. graph.) Velociy and Diplacemen (Honor) v (m/) Velociy v. Time area = diplacemen = (.5)(3 )(30 m/) + (4 )(30 m/) + (.5)(1 )(30 m/) = 180 m A velociy graph can be ued o deermine he diplacemen (change in poiion) of an objec. The area of he velociy graph equal he objec diplacemen. For a non-linear velociy graph, he area can be deermined by adding up infiniely many piece each of infiniely mall area, reuling in a finie oal area! Thi proce i now known a inegraion, and he funcion i called an inegral. 5
6 An Acceleraion Analogy Compare he graph of wage veru o a velociy veru graph. The lope of he wage graph i wage change rae. Slope of he velociy graph i acceleraion. Wha i he lope for each graph, including uni? In hi cae he wage change rae i conan. The graph i linear becaue he rae a which he wage change i ielf unchanging (conan)! The analogy help diinguih velociy from acceleraion becaue i i clear ha wage and wage change rae (acceleraion) are differen. lope = wage change rae = $1//hr/monh lope = acceleraion = 1 m// An Acceleraion Analogy Earning, Wage, and Wage Change Rae Can a peron have a high wage, bu a low wage change rae? Making good hourly money, bu geing very mall raie over. Can a peron have a low wage, bu a high wage change rae? Making lile per hour, bu geing very large raie quickly over. Can a peron have a poiive wage, bu a negaive wage change rae? Making money, bu geing cu in wage over. Can a peron have zero wage, bu ill have wage change rae? Making no money (inernhip?), bu evenually working for money. Poiion, Velociy, and Acceleraion Can an objec have a high velociy, bu a low acceleraion? Moving fa, bu only geing a lile faer over. Can an objec have a low velociy, bu a high acceleraion? Moving lowly, bu geing a lo faer quickly over. Can an objec have a poiive velociy, bu a negaive acceleraion? Moving forward, bu lowing down over. Can an objec have zero velociy, bu ill have acceleraion? A re for an inan, bu hen immediaely beginning o move. 6
7 Direcion of Velociy and Acceleraion v i a moion Velociy v. Time + 0 conan poiive vel. 0 conan negaive vel. 0 + peeding up from re 0 peeding up from re + + peeding up peeding up + lowing down + lowing down click for apple Kinemaic Equaion of Moion Auming conan acceleraion, everal equaion can be derived and ued o olve moion problem algebraically. v (m/) Velociy v. Time (Conan Acceleraion) Slope equal acceleraion a = Δv = v f v i v f = v i + a Area equal diplacemen v f A = 1 b 2 ( 1 + b 2 )h Δx = 1 v 2 ( + v i f ) v i Eliminae final velociy Δx = v i a 2 Eliminae v f 2 = v i 2 + 2aΔx 7
8 Freefall Acceleraion Ariole wrongly aumed ha an objec fall a a rae proporional o i weigh. Galileo aumed all objec freefall (in a vacuum, no air reiance) a he ame rae. An inclined plane reduced he effec of graviy, howing ha he diplacemen of an objec i proporional o he quare of. Δy 2 Since he acceleraion i conan, velociy i proporional o. v click for video Locaion g Equaor Honolulu Denver San Francico Munich Leningrad Norh Pole Laiude, aliude, geology affec g. Kinemaic equaion of freefall acceleraion: Δy = v yi g 2 v yf = v yi + g v yf 2 = v yi 2 + 2gΔy Δy = 1 v 2 ( yi + v yf ) 8
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