UNIT 1 ONE-DIMENSIONAL MOTION GRAPHING AND MATHEMATICAL MODELING Objeces To learn abou hree ways ha a physcs can descrbe moon along a sragh lne words, graphs, and mahemacal modelng. To acqure an nue undersandng o poson, elocy, and acceleraon or one-dmensonal moon. To recognze how graphs can be used o descrbe changes n poson, elocy and acceleraon o an objec mong along a sragh lne. To be able o use mahemacal denons o aerage elocy and aerage acceleraon n one dmenson. To check he aldy o some o he sandard knemacs equaons used o descrbe he moon o objecs undergong consan acceleraon. To be able o use knemacs equaons o descrbe he moon o objecs mong wh consan acceleraon. Equpmen ls: Maskng ape 4 Sopwaches 1 Bowlng Ball Ecel spreadshee 1.1 Below are a seres o snapshos o hree deren bowlng balls a mes 1,, 3, 4, and 5. The mes are equally spaced. Suppose each o he me nerals s 0.5s. 1
a. Deermne each bowlng ball s speedng up, slowng down, or mong a consan speed. Eplan your reasonng. b. In he rs me neral, compare he dsances coered by each bowlng ball. How does he dsance coered n ha me neral ge you normaon abou he speed o he bowlng ball? Eplan. 1. Oban a bowlng ball. Wh a pece o maskng ape, mark o a sarng pon, and dsances o m, 4m, 6m, and 8m rom he sarng pon. a. Predc wha wll happen, you release he bowlng ball a he sarng pon. Wll moe a consan speed, speed up or slow down? I you record he me reaches he m, 4m, 6m, and 8m marks, wll he me nerals beween adjacen marks ncrease, decrease or reman he same? Eplan. b. Perorm he epermen. Release he bowlng ball a he sarng pon. Use sopwaches, sarng hem a he me he ball s released, and recordng he me he ball crosses he m, 4m, 6m, and 8m marks. Record he posons and mes n an Ecel spreadshee. c. I you were o plo he poson on he ercal as and he me on he horzonal as, wha would he plo look lke? (Ths s a predcon.) d. In Ecel, use Graph o plo he poson on he ercal as and he me on he horzonal as. Eplan why looks he way does. How does compare o your predcon? 3
e. Wha would a graph o speed s. me look lke? Dscuss wh an nsrucor. Equpmen ls: Moon deecor LabPro compuer nerace LoggerPro soware.1 An ulrasonc moon deecor s a dece ha can be used o record an objec s moon as a uncon o me. I sends ou a seres o sound pulses (oo hgh requency o hear). These pulses relec rom objecs n he cny and he releced pulses reurn o he sensor. The compuer can record he me akes he releced pulses o reurn o he compuer, and usng he speed o sound, calculae he poson o he objec. When usng a moon sensor: 1. Do no ge closer han 0.5 meers rom he sensor because canno record releced pulses ha come back oo soon aer hey are sen.. The ulrasonc waes spread ou n a cone o abou 15 as hey rael. They wll see he closes objec. Be sure here s a clear pah beween he objec whose moon you wan o rack and he moon sensor. 3. The moon sensor s ery sense and wll deec slgh moons. You can ry o glde smoohly along he loor, bu don be surprsed o see small bumps n poson graphs and een larger bumps laer n elocy and acceleraon graphs. 4. Some objecs lke bulky sweaers are good sound absorbers and may no be seen ery well by a moon sensor. You may wan o hold a book n ron o you you hae loose clohng on. a. Predc and skech poson s. me (me on he horzonal as, poson on he ercal as) graphs or he ollowng suaons: sandng sll sarng near he deecor, walkng a a as seady walkng speed away rom he deecor sandng abou 3m away rom he deecor, walkng a a slow seady walkng speed owards he deecor, soppng brely, walkng a a as seady walkng speed away rom he deecor walkng a a slow seady walkng speed away rom he deecor, hen walkng a a as seady walkng speed urher away rom he deecor walkng away rom he deecor slowly, soppng or a whle, walkng aser away rom he deecor b. A your able you hae Moon Deecor, a LabPro compuer nerace and he manuals ha accompany ha equpmen. Connec he whe (USB) cable o he compuer and he oher end o LabPro compuer nerace (See he Verner LabPro User s Gude or ask 4
your nsrucor, you hae quesons.) Connec he power cord o he AC adaper por on he LabPro compuer nerace and plug n. Connec he gray Brsh Telecom end o he cable no DIG/SONIC on he LabPro nerace. Se he moon deecor where you hae an asle n ron o he deecor abou 5m long and 1.5m wde. Clck on hs lnk o he le Poson, go o Seup, choose Daa collecon, under Samplng se epermen o abou 3 sec. (You can change hs. You wll probably wan beween o 6 seconds, dependng on wha you are dong.) and se he samples per second o abou 40. Clck OK. Perorm he moons descrbed n par a. n ron o he moon deecor whle akng daa. c. Dscuss wh your parners he derence n your moon ha produces derenly sloped pars o he graph. Dscuss mong orward and backward and how aecs he graph.. Consder he ollowng graph. a. Descrbe n words how you would moe n ron o he moon deecor n order o creae he graph. b. Walk n ron o he moon deecor and ry o creae he graph..3 Consder he ollowng dagram. (rom Workshop Physcs (Elecronc Verson) by Prslla Laws, e al., John Wley and Sons, NY, 1999) a. Descrbe n words how you would moe n ron o he moon deecor n order o creae he graph. b. Walk n ron o he moon deecor and ry o creae he graph. 5
c. How s he graph deren rom he preous graph? How do you hae o walk derenly o creae he graph compared o how you had o walk o creae he preous graph? Dscuss wh an nsrucor. Equpmen ls: None 3.1 Consder he ollowng dagram. a. Dscuss he speed n each segmen o he graph. Descrbe n words how you would deermne a alue or he speed n each segmen o he graph. b. Are you able o deermne he drecon o moon n each segmen n he graph? Could you use a symbol o ndcae he drecon o moon? Eplan. How would you epress boh he speed and he drecon numercally? Dscuss he aboe wh an nsrucor. A quany ha epresses boh speed and drecon s called elocy. The denon o aerage elocy oer a me neral s he dsplacemen durng he me neral dded by he me neral. The dsplacemen s he nal poson mnus he nal poson and he me neral s he nal me mnus he nal me. We can wre hs mahemacally as = or, equalenly, = s he nal poson and s he nal poson. s he nal me and s he nal me. The uns o elocy are meers per second (m/s). Snce dsplacemen s a ecor quany, elocy s also a ecor quany. I has boh a magnude and a drecon. The magnude o he elocy s called he speed. 6
When workng n one dmenson, he drecon o he elocy can be descrbed as orward or backward, le or rgh, pose or negae, ec. We wll use he words pose or negae or he symbols + or o descrbe he drecon o he elocy. I s common, when workng n one dmenson, o wre he aboe equaon as a scalar equaon, where he aerage elocy s eher a pose or negae quany dependng on he drecon o moon =. c. Use he equaon or elocy o calculae he aerage elocy or each segmen o he graph. Record your resuls below. Tme Ineral (s) Speed (m/s) Drecon Velocy (m/s) 0-1.5 1.5-3 3-4.5 4.5-6 6-7 d. Deermne he aerage elocy oer he me neral rom 0-4.5s (boh he magnude and he drecon). Deermne he aerage elocy oer he me neral rom 0-7s (boh he magnude and he drecon). Check your calculaons wh an nsrucor. e. Make a graph o elocy s. me (elocy on he ercal as and me on he horzonal as) or he daa n par c. Equpmen Ls: Moon deecor LabPro Compuer Inerace LoggerPro soware 4.1 a. Predc and skech elocy s. me (me on he horzonal as, elocy on he ercal as) graphs or he ollowng suaons: sandng sll sarng near he deecor, walkng a slow speed away rom he deecor sarng away rom he deecor, walkng a a as speed owards he deecor sandng 3m away rom he deecor, walkng a a slow consan speed owards he deecor, soppng brely, walkng a a as consan speed away rom he deecor 7
walkng a a slow consan speed away rom he deecor, hen walkng a a as consan speed urher away rom he deecor b. Clck on hs lnk o he le Poson & elocy. The graph o elocy s. me wll be labeled elocy on he y-as. The moon deecor s acually measurng your elocy. Perorm he moons descrbed n par a. n ron o he moon deecor whle akng daa o es your predcons. 4. Consder he ollowng graph o elocy s. me. a. Descrbe n words how you would moe n ron o he moon deecor n order o creae he graph. b. Walk n ron o he moon deecor and ry o creae he graph. c. Draw he poson s. me graph ha corresponds o he moon o he elocy graph. Check your graph wh an nsrucor. 4.3 Consder he ollowng poson graphs. a. Calculae he slope o each segmen o he graph. b. Compare he slopes calculaed n par a o he aerage elocy n each o he me segmens n par 3.1.c. How do he slopes and he aerage eloces compare? c. Dscuss he saemen: The elocy s he rae o change o poson wh me. Dscuss he saemen: The elocy s he slope o a poson s. me graph. 8
4.4 a. Wha would he elocy s. me graph look lke, you were o moe, no a a consan speed, bu geng aser and aser and aser? (Ths s a predcon. Draw he graph.) Wha you were mong slower and slower and slower? b. I you were o moe aser and aser and aser a a consan rae (your elocy s ncreasng a a consan rae), wha would he elocy s. me graph look lke? (Ths s a predcon. Draw he graph.) c. Predc he poson s. me graph or par b. d. Se up he moon deecor and es your predcons o pars a and b. Equpmen Ls: Moon deecor LabPro Compuer Inerace LoggerPro soware Low Frcon Dynamc Track Car Car wh an accessory Rod Sand 5.1 Tl a Dynamcs rack a a small angle o he able, as n he dagram below. a. Place a car a he op o he rack and release. Plo he elocy s. me or he car, usng he moon deecor. Is he elocy ncreasng a a consan rae? Eplan how you know. b. Ge a denon o acceleraon. I does no hae o be ormal. (Descrbe your undersandng o wha means o accelerae.) c. Was he car n par 5.1.a accelerang? Eplan. The denon o aerage acceleraon oer a me neral s he change n elocy durng he me neral dded by he me neral. The change n elocy s he nal elocy mnus he nal elocy and he me neral s he nal me mnus he nal me. We can wre hs mahemacally as 9
a = or, equalenly, a = where s he nal elocy and s he nal elocy, s he nal me and s he nal me. The uns o acceleraon are meers per second per second (m/s ). Acceleraon s a ecor quany. I has boh a magnude and a drecon. As or elocy, when workng n one dmenson, he drecon o he acceleraon can be descrbed as orward or backward, le or rgh, pose or negae, ec. We wll use he words pose or negae or he symbols + or o descrbe he drecon o he acceleraon. I s common, when workng n one dmenson, o wre he aboe equaon as a scalar equaon, where he aerage acceleraon s eher a pose or negae quany dependng on he drecon o moon a =. d. Deermne he aerage acceleraon o he car n par a. Check your calculaon wh an nsrucor. 5. a. In each o he our cases below, deermne wheher he car s accelerang a car on a la low rcon dynamcs rack, whch has been gen an nal push (Deermne s accelerang aer he nal push. ) a car on a led rack, released rom he op o he rack beore hs he boom bumper, as n par 5.1.a. a car on a led rack, as n par 5.1.a, ha has been released rom he op o he rack, allowed o h he boom bumper and connue back up he rack (consder he moon aer has h he bumper and s mong up he rack) a car wh a an (oban hs rom an nsrucor) on op o wh he an runnng, on a la low rcon dynamcs rack b. Clck on hs lnk o open he le poson, elocy, and acceleraon and use he moon deecor o plo he acceleraon s. me or each o he cases n par 5..a. Is he daa conssen wh your answers o 5..a? c. Consder he ollowng dagram. 10
() Calculae he aerage acceleraon or each segmen. () Calculae he slope o he lne or each segmen. d. Dscuss he saemen: The acceleraon s he rae o change o elocy wh me. Dscuss he saemen: The acceleraon s he slope o a elocy s. me graph. e. Use he moon deecor o plo poson s. me, elocy s. me, and acceleraon s. me or a car on a led (a a small angle) low rcon dynamcs rack or a me long enough or he car o bounce o he bumper hree mes. Dscuss how he acceleraon s he slope o he elocy s. me graph and he elocy s he slope o he poson s. me graph or each segmen. Dscuss he graphs wh an nsrucor. Equpmen Ls: Moon deecor LabPro Compuer Inerace LoggerPro soware Low Frcon Dynamc Track Car Rod Sand Book Soball Bo Coee ler Alumnum Fol Super Ball 6.1 Consder he ollowng graph o acceleraon s. me. 11
a. For he me neral s 6s, wha s he aerage acceleraon? Eplan. Would he aerage acceleraon be deren or a deren me neral? Eplan. b. I he acceleraon o an objec remans consan, wha saemen can you make abou he aerage acceleraon? Eplan. c. Descrbe any cases you can hnk o where he acceleraon o a mong objec remans consan. 6. Hang a moon deecor rom he celng wh he sde wh he sensor owards he loor. a. Open he poson, elocy, and acceleraon le and se he epermen lengh or 1s. Use he moon deecor o plo he acceleraon s. me or each o he ollowng cases: () Drop a book rom 0.5m below he deecor o he loor. () Drop a soball rom 0.5m below he deecor o he loor. () Drop a bo rom 0.5m below he deecor o he loor. b. Compare he acceleraons o he hree objecs n par 6..a. For each objec, s he acceleraon appromaely consan whle he objec s allng? Eplan. c. Do all objecs allng owards he earh all wh he same acceleraon? Eplan. d. Use he moon deecor o plo he acceleraon s. me or each o he ollowng cases: () Drop a coee ler rom 0.5m below he deecor o he loor. () Drop a pece o alumnum ol rom 0.5m below he deecor o he loor. () Drop a super ball rom 0.5m below he deecor o he loor. e. For whch o he objecs you hae obsered n pars a and d s here ery lle ar rcon? For whch o he objecs you hae obsered n pars a and d s here a lo o ar rcon? 1
Eense epermens hae been done ha ndcae ha all objecs all owards he earh wh a consan acceleraon ar rcon s neglgble (here s ery lle ar rcon). The magnude o ha acceleraon s appromaely 9.8m/s near he surace o he Earh. The symbol g s oen used o represen he magnude o hs acceleraon (g = 9.8m/s ). I you dd no ge good daa or he super ball, look a hs le. Take noe a he acceleraon, on aerage, s 9.8m/s.. For he objecs you obsered n pars a and d or whch here was ery lle ar rcon, how does he acceleraon compare o he alue 9.8m/s? 6.3 Use he moon deecor o plo he acceleraon s. me or a car on a low rcon dynamcs rack led a a small angle. a. Whle he car s on he rack, ecep when hs he bumper, s he acceleraon consan? Eplan. b. When he acceleraon s consan, descrbe he graph o elocy s. me. For he graph n 6.1.a, draw a possble graph o elocy s. me. Dscuss your answer wh an nsrucor. c. I can be shown, ha or a sragh lne graph ha he aerage alue beween wo pons on he lne s he alue o he pon halway beween he wo pons. For eample, or a plo o elocy s. me, he wo pons chosen are 1 and, as shown n he dagram, + below, he aerage alue oer he me neral 1 o s =, as shown. Very he equaon = +, usng he elocy s. me graph you generaed n 6.3.a. Snce he elocy s. me graph s always a sragh lne or he case o consan + acceleraon, he equaon = s always rue or consan acceleraon. Equpmen Ls: 13
Moon deecor LabPro Compuer Inerace LoggerPro soware Low Frcon Dynamc Track Car Car wh an accessory Rod Sand Meer Sck 7.1 For objecs mong a consan acceleraon, he hree equaons below can be appled. = a = + = I s also possble o manpulae hese hree equaons o oban our knemacs equaons or use n descrbng objecs mong a consan acceleraon. These equaons are gen below. ( ) ( ) ( ) ( ) ( ) - a a 1 - a - = + = = + = where, he nal me mnus he nal me, s he me neral, (he nal poson mnus he nal poson) s he dsance raeled, and are he nal and nal eloces and ā s he acceleraon. These equaons are somemes wren as 14
where = s he me neral. + - = = a a - ( - ) = = + These equaons are ald only or he case o consan acceleraon. a. Se up an epermen wh consan acceleraon ha wll allow you o measure poson, elocy and acceleraon as a uncon o me wh he moon deecor and ery he our knemacs equaons gen aboe. b. Hae someone hold a meer sck, as he person on he rgh n he pcure below. A second person places hs/her hands near he boom o he meer sck wh humb and orenger so ha hey can cach he meer sck when alls. Record he poson o he second person s humb and orenger (wha number on he meer sck hey are algned wh). Hae he person holdng he meer sck release he meer sck (The person should jus release all o a sudden.). The second person should cach as soon as hey percee s released. Record he poson o he person s humb and orenger aer hey hae caugh he meer sck. Deermne he dsance he meer sck ell. Use one o he knemacs equaons o deermne he person s reacon me. Deermne he reacon me or each person n your group. 1 a (rom Fundamenals o College Physcs nd Edon by Peer J. Nolan, Wm. C. Brown Publshers, Chcago, 1995) 15
SUMMARY You should be able o descrbe n words he moon o an objec ha would produce a parcular graph o poson, elocy or acceleraon s. me. Gen he moon o an objec, you should be able o draw he poson, elocy, and acceleraon graphs. Gen a parcular graph (eher poson, elocy or acceleraon), you should be able o draw oher graphs: or eample, gen a poson graph, you should be able o draw he elocy and acceleraon graphs. You should undersand he mahemacal denon o aerage elocy, be able o use, and undersand ha elocy s he rae o change o poson and can be ound rom he slope o he poson graph. You should undersand he mahemacal denon o aerage acceleraon, be able o use, and undersand ha acceleraon s he rae o change o elocy and can be ound rom he slope o he elocy graph. You should be able o use he knemacs equaons or he case o consan acceleraon o descrbe he moon o mong objecs and o sole or unknown parameers. 16