CHAPTER Decribing Min: Kinemaic in One Dimenin hp://www.phyicclarm.cm/cla/dkin/dkintoc.hml Reference Frame and Diplacemen Average Velciy Inananeu Velciy Accelerain Min a Cnan Accelerain Slving Prblem Falling Objec Graphical Analyi f Linear Min DISPLACEMENT There are w apec any min. There i he mvemen ielf and here i he iue f wha caued he min r wha changed i. Thi require ha frce be cnidered. Kinemaic deal wih he cncep ha are needed decribe min, wihu any reference f frce. Dynamic deal wih he effec ha frce have n min. Tgeher, kinemaic and dynamic frm he branch f phyic knwn a mechanic. Quaniie in Min Any min invlve hree cncep Diplacemen Velciy Accelerain Thee cncep can be ued udy bjec in min T decribe he min f an bjec i lcain mu be knwn a all ime. Piin Defined in erm f a frame f reference One dimeninal, generally he x- r y-axi Define a aring pin fr he min
Diplacemen Diplacemen i a vecr ha pin frm an bjec iniial piin i final piin and ha a magniude ha equal he hre diance beween he w piin. The SI uni fr diplacemen i he meer (m). Scalar are quaniie ha are fully decribed by a magniude (r numerical value) alne. Vecr are quaniie ha are fully decribed by bh a magniude and a direcin. Generally dened by bldfaced ype and an arrw ver he leer + r ign i ufficien fr hi chaper _ x x x f i f and fr final and i and fr iniial May be repreened a y if verical Uni are meer (m) in SI, cenimeer (cm) in cg r fee (f) in US Cumary Example f Piin, Diance, and Diplacemen The diance i he al lengh f ravel; if yu drive frm yur hue he grcery re and back, yu have cvered a diance f 8.6 mi. Diplacemen i he change in piin. If yu drive frm yur hue he grcery re and hen yur friend hue, yur diplacemen i. mi and he diance yu have raveled i 0.7 mi. Diplacemen In Diance The diplacemen f an bjec i n he ame a he diance i ravel Example: Thrw a ball raigh up and hen cach i a he ame pin yu releaed i The diance i wice he heigh The diplacemen i zer
Speed The average peed f an bjec i defined a he al diance raveled divided by he al ime elaped. If a mrcycle ravel 300 meer in 0 ecnd he average peed i 30 m/. The average peed i he diance raveled divided by he ime required cver he diance raveled divided by he ime required cver he diance. Speed i a calar quaniy Average peed ally ignre any variain in he bjec acual min during he rip The al diance and he al ime are all ha i impran SI uni are m/ Velciy I ake ime fr an bjec underg a diplacemen The average velciy i rae a which he diplacemen ccur generally ue a ime inerval, le i = 0 Direcin will be he ame a he direcin f he diplacemen (ime inerval i alway piive) + r - i ufficien Uni f velciy are m/ (SI), cm/ (cg) r f/ (US Cu.) Oher uni may be given in a prblem, bu generally will need be cnvered hee Speed v. Velciy v al diance Average peed al ime v average d x xf xi f i Car n bh pah have he ame average velciy ince hey had he ame diplacemen in he ame ime inerval The car n he blue pah will have a greaer average peed ince he diance i raveled i larger 3
Graphical Inerpreain f Velciy Velciy can be deermined frm a piin-ime graph Average velciy equal he lpe f he line jining he iniial and final piin An bjec mving wih a cnan velciy will have a graph ha i a raigh line Average Velciy, Cnan The raigh line indicae cnan velciy The lpe f he line i he value f he average velciy Example : The piin f a runner a a funcin f ime i pled a mving alng he x axi f a crdinae yem. During a 3.00 ime inerval, he runner piin change frm x 50m x 30.5m. Wha wa he runner average velciy? given find frmula luin x 50.0m v x x 9.5m v v 6.50 m / x 30.5m 3.00 3.00 (Ne: diance (x) i he lef n he x-axi ha i why i i negaive) Example : Yu ep n a nail wih yur bare fee. A nerve impule, generaed in yur f, ravel hrugh yur nervu yem a an average peed f 0 m/. Hw much ime de i ake fr he impule, which ravel a diance f.8 m, reach yur brain? given find frmula luin v = 0 m/ x.8m 0.06ec v 0 m/ x =.8 m Example 3: A plane i iing n a runway, awaiing akeff. On an adjacen parallel runway, anher plane land and pae he ainary plane a a peed f 45 m/. The arriving plane ha a lengh f 36 m. By lking u f a windw, a paenger n he ainary plane can ee he mving plane. Fr hw lng a ime i he mving plane viible? given find frmula luin v = 45 m/ x 36m 0.8ec v 45 m/ x = 36 m 4
Example 4: Hw far can a cycli ravel in.5h alng a raigh rad if her average velciy i 8km/h? given find frmula luin =.5h x x v x (8 km / h)(.5 h) 45km v = 8km/h Inananeu Velciy The limi f he average velciy a he ime inerval becme infinieimally hr, r a he ime inerval apprache zer v lim 0 x The inananeu velciy indicae wha i happening a every pin f ime Thi pl hw he average velciy being meaured ver hrer and hrer inerval. The inananeu velciy i angen he curve. Example 5: Obain average and inananeu velciie frm a graph. A rain mve lwly alng a raigh rack (ee graph). Find: (a) he average velciy fr he al rip. Find he average velciy frm he rigin he final pin c. Calculae he lpe f he dahed blue line: x 0.0m v m r m.0 0.833 / ( 0.833 ) (b) The average velciy during he fir 4.00 f min. x 4.00m v.00 m / ( r.00 m ) 4.00 (c) The average velciy during he nex x 0m v m r m 4.00 0 / ( 0 ) 4.00 f min. (d) The inananeu velciy a.00. v.00 m / Thi i he ame a he average velciy fund in (b), becaue he graph i a raigh line. 5
(e) The inananeu velciy a = 9.00. x 4.5m 0m v 0.75 m / 9.0 3.0 The angen line appear inercep he x-axi a (3.0, 0m) and graze he curve a (9.0, 4.5m). The inananeu velciy a = 9.00 equal he lpe f he angen line hrugh hee pin. Accelerain Changing velciy (nn-unifrm) mean an accelerain i preen Accelerain i he rae f change f he velciy Uni are m/² (SI), cm/² (cg), and f/² (US Cu) Average Accelerain Vecr quaniy When he ign f he velciy and he accelerain are he ame (eiher piive r negaive), hen he peed i increaing When he ign f he velciy and he accelerain are in he ppie direcin, he peed i decreaing Average accelerain: v vf vi a f i Typical Accelerain (m/ ) Ulracenrifuge Bulle fired frm a rifle Baed baeball Airbag deplymen Bungee jump High jump Graviy n Earh Emergency p in a car Airplane during akeff An elevar Graviy n he Mn 3 x 0 6 4.4 x 0 5 3 x 0 4 60 30 5 9.80 8 5 3.6 Accelerain i he rae f change f velciy. 6
Accelerain i a vecr, alhugh in ne-dimeninal min we nly need he ign. The previu image hw piive accelerain; here i negaive accelerain: There i a difference beween negaive accelerain and decelerain: Negaive accelerain i accelerain in he negaive direcin a defined by he crdinae yem. Decelerain ccur when he accelerain i ppie in direcin he velciy. Graphical Inerpreain f Accelerain Average accelerain i he lpe f he line cnnecing he iniial and final velciie n a velciy-ime graph Inananeu accelerain i he lpe f he angen he curve f he velciy-ime graph Relainhip beween Accelerain and Velciy Unifrm velciy (hwn by red arrw mainaining he ame ize) Accelerain equal zer 7
Velciy and accelerain are in he ame direcin Accelerain i unifrm (blue arrw mainain he ame lengh) Velciy i increaing (red arrw are geing lnger) Piive velciy and piive accelerain Accelerain and velciy are in ppie direcin Accelerain i unifrm (blue arrw mainain he ame lengh) Velciy i decreaing (red arrw are geing hrer) Velciy i piive and accelerain i negaive Example 6: A car accelerae alng a raigh rad frm re 75 km/h in 5.0. Wha i he magniude f i average accelerain? given find frmula luin a v v 0.8 m / 0 m / v 0 m/ a a 4.6 m / 5.0 0 v 75 km / h 0.8 m/ (Ne: Cnver km/h m/ = dividing km/h by 3.6 will give yu m/) Example 7: A car i mving alng a raigh rad. The driver pu n he brake when he iniial velciy i 5.0 m/, and i ake 5.0 lw dwn 5.0 m/. Wha wa he car average accelerain? given find frmula luin v 5.0 m/ v 5.0 m/ 5.0 a v v a 5.0 m/ 5.0 m/ a.0 m/ 5.0 Example 8: A baeball player run he ufield. Hi velciy a a funcin f ime i hwn in he graph. Find hi inananeu accelerain a pin A, B, C. A each pin, he velciy v. ime graph i a raigh line egmen inananeu accelerain will be he lpe. Accelerain a A: The accelerain equal he lpe f he line cnnecing he pin (0, 0m/) and.0, 4.0m/) v 4.0 m / 0 a.00.0 m/ 8
Accelerain a B: v 0, becaue he egmen i hriznal v 4.0 m / 4.0 m / a 3.0.0 0 m/ Accelerain a C: The accelerain equal he lpe f he line cnnecing he pin (3.0, 4.0m/) and (4.0,.0m/) v.0 m/ 4.0 m/ a.0 m/ 4.0 3.0 Kinemaic Equain Ued in iuain wih unifrm accelerain (THESE ARE KEY EQUATIONS) THE BIG SIX Miing variable Ne n he equain x v v v a x x v a x x v a v v a( x x ) x ( v v) v v a a x v v Shw velciy a a funcin f accelerain and ime Ue when yu dn knw and aren aked find he diplacemen x v a Give diplacemen a a funcin f ime, velciy and accelerain Ue when yu dn knw and aren aked find he final velciy v v ax Give velciy a a funcin f accelerain and diplacemen Ue when yu dn knw and aren aked fr he ime 9
Prblem-Slving Hin Read he prblem Draw a diagram Che a crdinae yem, label iniial and final pin, indicae a piive direcin fr velciie and accelerain Label all quaniie, be ure all he uni are cnien Cnver if neceary Che he apprpriae kinemaic equain Slve fr he unknwn Yu may have lve w equain fr w unknwn Check yur reul Eimae and cmpare Check uni Example 9: Yu are deigning an airpr fr mall plane. One kind f plane mu reach a peed f 7.8 m/ befre akeff, and i able accelerae a.00. (a) mif / he runway i 50 m lng, can hi plane reach he required peed fr ake ff? given find frmula luin x v 0 0 x50m a.00 m / Thi runway lengh i n ufficien. (b) Wha minimum lengh mu he runway have? v v a( x x) v v (7.8 m/ ) 0 ( x x ) 93m a (.0 m / ) Example 0: Hw lng de i ake a car cr a 30.0 m wide ree afer he ligh urn green, if he car accelerae frm re a a cnan.00? m/ given find frmula luin x 30.0m a.00 m / v v v a( x x ) x a Example : Airbag can prec driver in a head-n clliin a a peed f 00km/h. Eimae hw fa he air bag mu inflae prec he driver. (diance car crumple i m) given find frmula luin v 00 km / h.8 m/ a v v (8 m/ ) v 0 a a x 390 m/ x.0m x m v v v v 0 8 m/ 0.07 a a 390 m / T be effecive, he air bag wuld need inflae faer han hi. x a 0 (.0 m/ )(50 m) 600 m / v 600 m / 4.5 m / x (30.0 m) 5.48 a.00 m / 0
Example : A race car aring frm re accelerae a a cnan rae f velciy f he car afer i ha raveled 30.5m? 5.00 m/. Wha i he given find frmula luin a = 5.00m/ x = 30.5 m v v v ax v v ax v m m (0) (5.00 / )(30.5 ) 7.5 m/ Example 3: A je aking ff frm he deck f an aircraf carrier ar frm re and i caapuled wih a cnan accelerain f 3 m/ alng a raigh line. The je reache a velciy f 6 m/. Find he diplacemen f he je. given find frmula luin a = 3 m/ x v v a( x x ) v = 6 m/ v v x (6 m/ ) 0 x 6m a (3 m/ ) Example 4: Fr a andard prducin car, he highe rad-eed accelerain ever repred ccurred in 993, when a Frd RS00. Evluin wen frm zer 6.8 m/ in 3.75. Find he magniude f he car accelerain. given find frmula luin v = 0 m/ a v = 6.8 m/ v v 6.8 m/ 0 v v a a a 8.8 m / 3.75 = 3.75 Galile Galilei 564-64 Galile frmulaed he law ha gvern he min f bjec in free fall Al lked a: Inclined plane Relaive min Thermmeer Pendulum Free Fall All bjec mving under he influence f graviy nly are aid be in free fall Free fall de n depend n he bjec riginal min All bjec falling near he earh urface fall wih a cnan accelerain The accelerain i called he accelerain due graviy, and indicaed by g.
Symblized by g g = 9.80 m/² Accelerain due Graviy When eimaing, ue g» 0 m/ g i alway direced dwnward ward he cener f he earh Ignring air reiance and auming g den vary wih aliude ver hr verical diance, free fall i cnanly acceleraed min Falling Objec Near he urface f he Earh, all bjec experience apprximaely he ame accelerain due graviy. Thi i ne f he m cmmn example f min wih cnan accelerain. Free fall i he min f an bjec ubjec nly he influence f graviy. The accelerain due graviy i a cnan, g. Freely Falling Objec In he abence f air reiance, all bjec fall wih he ame accelerain, alhugh hi may be hard ell by eing in an envirnmen where here i air reiance. Free Fall an bjec drpped Iniial velciy i zer Le up be negaive Ue he kinemaic equain Generally ue y inead f x ince verical Accelerain i g = 9.80 m/
Example 5: A ball i drpped frm a wer 70 m high. Hw far will he ball have fallen afer.00,.00, and 3.00? given find frmula a 9.80 m / y y v a y 0 (9.80 m / )(.00 ) 4.90m,,&3 y a (9.80 m / )(.00 ) 9.6 m a = g = 9.80 m/ Iniial velciy 0 iniial velciy will be piive Free Fall an bjec hrwn dwnward Example 6: Suppe he ball in he previu example i hrwn dwnward wih an iniial velciy f 3.00 m/, inead f being drpped. (a) Wha hen wuld be i piin afer.00 and.00? y v a (3.00 m / )(.00 ) (9.80 m / )(.00 ) 7.90m (3.00 m / )(.00 ) (9.80 m/ )(.00 ) 5.6m (b) Wha wuld i peed be afer.00 and.00? v v a 3.00 / (9.80 / )(.00 ).8 / m m m 3.00 m/ (9.80 m/ )(.00 ).6 m/ (9.80 / )(3.00 ) 44. y3 a m m Example 7: A pern ep ff he end f a 5.00-m-high diving plafrm and drp he waer belw. Hw lng de i ake fr he pern each he waer belw? given find frmula luin x x = 5.0 m x x v a 0 0 g g (5.00 m) g = 9.80 m/.0ec 9.80 m/ Free Fall -- bjec hrwn upward Iniial velciy i upward, piive The inananeu velciy a he maximum heigh i zer a = g = -9.80 m/ everywhere in he min 3
The min may be ymmerical Then up = dwn Then v = -v The min may n be ymmerical Break he min in variu par Generally up and dwn Example 8: A pern hrw a ball upward in he air wih an iniial velciy f 5.0m/. Calculae (a) Hw high i ge v v 0 (5.0 m/ ) y.5m a ( 9.80 m / ) (b) Hw lng he ball i in he air befre i cme back he hand? y v a 0 (5.0 m / ) ( 9.80 m / ) (5.0 m/ 4.90 m/ ) 0 5.0 m/ 0 & 3.06 4.90 m/ (c) Hw lng de i ake he ball reach he maximum heigh? v v a v 5.0 m/.53 a 9.80 m / (d) Wha i he velciy f he ball when i reurn he hrwer hand? v v a 5.0 m/ (9.80 m/ )(3.06 ) 5.0 m/ (e) Calculae a wha ime he ball pae a pin 8.00 m abve he pern hand. y y v a 8.00m 0 (5.0 m / ) (9.80 m / ) b b 4ac a (4.90 m/ ) (5.0 m/ ) (8.00 m) 0 5.0 m / (5.0 m / ) 4(4.90 m / )(8.00 m) (4.90 m/ ) 0.69 and.37 4
Nn-ymmerical Free Fall Need divide he min in egmen Pibiliie include Upward and dwnward prin The ymmerical prin back he releae pin and hen he nn-ymmerical prin Example 9. A lava bmb i launched raigh up frm a vlcan. Uing a pwach i i deermined ha i k 4.80 rie up and land n he grund. Accelerain wa 9.80 m/ dwnward. Wha wa he iniial peed? x 0 and a g x x v a v g (v g) x ( v g) 0 0 v g 0 v g 0 r v g (9.80 / v g m )(4.80 ) 3.5 m / Example 0. A car and a ruck are heading direcly ward ne anher n a raigh and narrw ree, bu hey avid a head-n clliin by imulaneuly applying heir brake a = 0. Wha i he eparain beween he car and he ruck when hey have cme re, given ha a = 0 he car i a x = 5m and he ruck i a x = -35m. Picure he Prblem: The velciy-veru-ime pl f he car and he ruck are hwn a righ. The car begin wih a piive piin and a negaive velciy, i mu be repreened by he lwer line. The ruck begin wih a negaive piin and a piive velciy, i i repreened by he upper line. Sraegy: The diance raveled by he car and he ruck are equal he area under heir velciy-veru-ime pl. We can deermine he diance raveled frm he pl and ue he knwn iniial piin find he final piin and he final eparain. 5
Sluin:. Find he final piin f he ruck: ruck 0,ruck ruck x x x 35 m.5 0 0 m/.5 m. Find he final piin f he car: xcar x0,car xcar 5 m 3.5 0 5 m/.5 m 3. Nw find he eparain: x x car ruck.5 m.5 m.3 m Example 0. A h-air balln lifed ff he grund and i riing a a rae f.0m/. One f he paenger realize he ha lef her camera n he grund. A friend pick i up and e i raigh upward wih an iniial peed f 3 m/. If he paenger i.5m abve her friend when he camera i ed, hw high i he when he camera reache her? Picure he Prblem: The rajecrie f he balln and camera are hwn a righ. The balln rie a a eady rae while he camera peed i cninually lwing dwn under he influence f graviy. The camera i caugh when he w rajecrie mee. Sraegy: The equain f min fr piin a a funcin f ime and velciy (equain -0) can be ued decribe he balln, while he equain fr piin a a funcin f ime and accelerain (equain -) can be ued decribe he camera min. Se hee w equain equal each her find he ime a which he camera i caugh. Then find he heigh f he balln a he inan he camera i caugh. Sluin:. Wrie equain -0 fr he balln: xb xb,0 vb. Wrie equain - fr he camera: x 0 v g c c,0 3. Se xb xc and lve fr : x v v g b,0 b c,0 0 x v v g b,0 c,0 b 4. Muliply by and iner he number: 0.5 m 3.0 m/ 9.8 m/ 5. Apply he quadraic frmula and lve fr. The larger r crrepnd he ime when he camera wuld pa he balln a ecnd ime, n i way dwn back he grund. 0.5 4.9 4 4.9.5 b b 4ac a 9.8 0.6 r.0 6. Find he heigh f he balln a ha ime: x x v,0.5 m.0 m/ 0.6 3.0 m b b b Inigh: If he paenger mie he camera he fir ime, he ha anher h a i afer.0 (frm he ime i i hrwn) when he camera i n i way back ward he grund. 6
CHAPTER MOTION IN ONE DIMENSION CONCEPTS. An bjec velciy can change direcin when i accelerain i cnan. An example f hi i when an bjec, like a ball, i hrwn raigh up. The velciy i piive ging up and negaive ging dwn. The accelerain will remain cnan he enire ime.. An bjec can have increaing peed while i accelerain i decreaing. An example wuld be an bjec releaed frm re in he preence f air fricin. 3. An bjec i mving wih cnan accelerain in equal ime i velciy change by equal amun. 4. When a ball i hrwn raigh up, i velciy a he p i zer. 5. When a ball i hrwn raigh up, i accelerain a he p i g (9.80 m/ ) 6. A ball, mve up alng a mh hill f ice will have he ame accelerain, bh up he hill and dwn he hill. 7. A kydiver jump frm an airplane. When he reache erminal velciy, her accelerain i eenially zer. 8. Objec A and B bh ar frm re. They bh accelerae a he ame rae. Hwever, bjec A accelerae fr wice he ime a bjec B. During he ime ha he bjec are being acceleraed, cmparing he diance raveled by bjec A ha f bjec B, bjec A ravel fur ime a far. 9. Objec A and B bh ar frm re. They bh accelerae a he ame rae. Hwever, bjec A accelerae fr wice he ime a bjec B. Cmpared he final peed f bjec A ha f bjec B, bjec B i wice a fa. 0. A ball i drpped frm he p f a building. A ecnd ball i hrwn raigh dwn frm he ame building. They are releaed a he ame ime. Neglecing air reiance he w ball accelerae a he ame rae.. Ball A i drpped frm he p f a building. One ecnd laer ball B i drpped. A ime prgree he difference in heir peed remain cnan.. Ball A i drpped frm he p f a building. One ecnd laer, ball B i drpped frm he ame building. A ime prgree, he diance beween hem increae. 3. Galile pulaed ha a a given lcain n he Earh and in he abence f air reiance, all bjec will fall wih he ame cnan accelerain. 7
4. Tw ball are hrwn raigh up. The fir ball i hrwn wih wice he iniial peed f he ecnd. Ignre air reiance. The fir ball will rie fur ime a far. 5. Tw bjec are hrwn frm he p f a all cliff. One i hrwn up, and he her i hrwn dwn, bh wih he ame iniial peed. Ignre air reiance, when he ball hi he grund hey are raveling he ame peed. 6. A hriznal line n a diance (piin) veru ime graph indicae ha he bjec i a re. 7. A mving bjec mu underg a change f piin. 8. A hriznal line n a velciy veru ime graph indicae ha he bjec i raveling a a cnan velciy, which mean zer accelerain. 9. An bjec wih zer accelerain may be in min. 0. A hriznal line n an accelerain veru ime graph indicae ha he bjec i a a cnan accelerain.. An bjec velciy can change direcin when i accelerain i cnan. Example, when a rck i hrwn raigh up.. The diance ime graph n he righ cmpare diance a a funcin f ime an bjec in raigh-line min. Accrding he graph, he bjec m likely ha an increaing velciy. 3. The diance ime graph n he righ repreen he min f an bjec liding dwn a fricinle inclined plane. 4. If an bjec i raveling ea wih a decreaing peed, he direcin f he bjec accelerain i we. 5. Galile pulaed ha all bjec will fall wih he ame cnan accelerain in he abence f air r her reiance. 6. The lpe f a piin ime graph give inananeu velciy. 7. The lpe f a velciy veru ime graph give inananeu accelerain. 8. Diance i diplacemen a peed i velciy. 9. If yu run arund a rack and reurn he ame aring pin, yur average velciy i zer. (Velciy ue diplacemen raher han diance. The diplacemen in hi cae i zer.) 30. Accelerain i a vecr quaniy ha repreen he ime-rae f change in velciy. 8
3. When an bjec i releaed frm re and fall in he abence f fricin i accelerain i cnan. 3. Inananeu peed i never negaive. 33. Min in he negaive x direcin i repreened n an x v. pl by a dwnward lping curve. 34. Oil drip a 0.5 ecnd inerval frm a ruck ha ha an il leak. The paern hwn repreen he pacing f il drp a he car accelerae unifrmly frm re. 9