13.1 Accelerating Objects

Transcription

1 13.1 Acceleraing Objec A you learned in Chaper 12, when you are ravelling a a conan peed in a raigh line, you have uniform moion. However, mo objec do no ravel a conan peed in a raigh line o hey do no have uniform moion. For example, a cycli peed up, change direcion, or doe boh while riding a bicycle (Figure 1). When an objec change peed or direcion, i i changing i velociy. Acceleraion i defined a he rae of change of velociy. Acceleraion i a vecor quaniy imilar o diplacemen and velociy. I herefore ha boh a magniude and direcion. The average acceleraion i he change of velociy divided by he ime inerval aken for he change. Thi i wrien a he equaion cha nge of ime in velociy erval v where i he average acceleraion in m/ 2, v he change in velociy in m/, and i he ime in. Noe ha he uni for acceleraion i mere per econd per econd (m//), which i abbreviaed o mere per econd quared (m/ 2 ). Since acceleraion i he change in velociy, which i calculaed by ubracing he iniial velociy (v ) from he final velociy (v f ), we can alo wrie he equaion a v LEARNING TIP Preview Chaper 13 o ge an overall impreion of he conen. Skim he heading, able, figure, and ample problem. Review he key idea and chaper preview. Wha informaion will be new o you? Wha opic do you need o focu your aenion on? To learn more abou how change in velociy relae o acceleraion, wach he animaion a Figure 1 Thee cycli in he Tour de France are ravelling a a conan peed, however, heir moion i acceleraed Acceleraing Objec 373

2 SAMPLE PROBLEM 1 Deermine Acceleraion A baeball i hrown oward a baer wih a velociy of 30.0 m/. The baer hi he ball wih he ba o ha he ball flie away a a velociy of 25.0 m/. The ball wa in conac wih he ba for Wha i he average acceleraion of he ball while being hi? Soluion Le he direcion away from he baer be poiive (and he direcion oward he baer be negaive). Subiue he value ino he acceleraion equaion. STUDY TIP Develop he udy kill of planning ahead. Plan o read each ecion before i i dicued in cla. Doing hi will help you underand he maerial ha i preened in cla a a deeper level. v f v 25.0 m/ (30.0 m/) m/ 2 The average acceleraion of he baeball while being hi i m/ 2, or m/ 2 away from he baer. Pracice A helicoper wa ravelling ea a 14 m/ (50 km/h) oberving raffic. The helicoper urned, and 35 laer i wa ravelling we a 14 m/. Wha wa he average acceleraion of he helicoper during hi period? An objec in uniform moion ha i ravelling a a conan velociy (peed and direcion are no changing) ha no acceleraion. For example, a car ravelling along a raigh road a a conan peed ha an acceleraion of zero. However, an objec ha i ravelling in a circle a a conan peed i coninuouly changing i direcion and, herefore, i undergoing acceleraion. Plane, Ferri wheel, and merry-go-round, have moion like hi. Since i i o common, hi ype of moion i called uniform circular moion. You will udy uniform circular moion in enior phyic coure. 13A Inveigaion Analyi of an Objec Moving a Conan Acceleraion To perform hi inveigaion, urn o page 393. In hi inveigaion, you will examine and analyze he moion of an objec a i accelerae from re. Acceleraion from Re An aricle in he newpaper ae ha a car can go from zero o 100 km/h in 3.2. Thi i a meaure of acceleraion. Thi could be wrien a v 100. km/h 0 km/h 31 km/h/ 3.2 We read hi a hiry-one kilomere per hour per econd. In oher word, he velociy forward increae by 31 km/h each econd. Therefore, alhough he bae meric uni for acceleraion i m/ 2, i i poible o ue oher uni o expre he rae of change of velociy. How can you deermine he acceleraion of an objec in he laboraory? Uing a ramp and a recording imer, you can inveigae he moion of an acceleraing objec. 13A Inveigaion 374 Uni D Moion

3 Falling Objec: Acceleraion Due o Graviy In he Try Thi aciviy a he beginning of hi chaper, you compared he rae a which differen objec fall o he ground. Objec fall becaue he force of graviy arac hem o Earh (Figure 2). For objec uch a marble and book, we can aume ha he effec of air reiance i negligible. When air reiance i negligible, he moion i known a free fall. Figure 2 Thi bungee jumper enjoy a brief momen of free fall before he elaic cord caue he jumper o rebound upward. For example, le look a an experimen ha udied he poiion and ime of a falling marble. The marble wa videoaped a i fell beide a mere ick. The daa given in Table 1 wa recorded by waching he video in low moion. Figure 3 how he poiion ime graph. Table 1 Poiion of a Falling Marble Poiion (cm down) Poiion (cm down) Moion of a Marble Figure 3 Poiion ime graph for he moion of a falling marble You have learned ha he lope of a line on a poiion ime graph equal velociy. We can calculae he lope of he wo angen o he graph o deermine he inananeou velociy of he marble a 0.1 and 0.3 : ri v 0.1 r u e n 2 5 cm 0 cm cm/ down rie v cm 25 cm 300 cm/ down r un Acceleraing Objec 375

4 Table 2 how he reul if he inananeou velociy i calculaed for oher ime. A graph of velociy ime i hown in Figure 4. Table 2 Velociy of a Falling Marble Velociy (cm/ down) Velociy (cm/ down) Velociy of a Marble Did You KNOW? Acceleraion of Graviy in Space The acceleraion of graviy on he urface of he Moon i 1.6 m/ 2 (abou one-ixh of ha on Earh) and on Mar i 3.7 m/ 2 (abou one-hird). The graviy on Nepune i 14.1 m/ 2, which mean ha i generae an acceleraion of graviy ha i almo 1.5 ime greaer han ha of Earh. To learn more abou free fall, go o GO Figure 4 Velociy ime graph for a falling marble We can ee from Figure 4 ha he velociy of he marble increaed wih ime. By drawing he line of be fi, we can ee ha he velociy increaed a a conan rae, alhough he incremen were no equal becaue here were ligh error in he daa. The rae ha he marble velociy increaed i he acceleraion of he marble and i caued by he force of graviy. Since we know ha he acceleraion of graviy i fairly conan, we know ha he acceleraion of he marble i alo conan. Uing hi definiion, we can calculae he free fall acceleraion of he marble a v 491 cm / 0 cm/ cm/ 2 down. 50 Thi value i he acceleraion caued by he force of graviy. The force of graviy change lighly from one locaion on Earh o anoher. The average value of acceleraion of graviy on Earh i 9.8 m/ 2 down. Noe ha hi value doe no apply o objec ha are affeced by air reiance. Negaive Acceleraion Acceleraion i defined a a change in velociy over ime. The change in velociy can be eiher an increae or a decreae. An objec ha undergoe a decreae in velociy, ha i, he final velociy i le han he iniial velociy, ha negaive acceleraion. Thi mean ha he change in velociy will be a negaive number. For example, a car i ravelling we (le hi be he poiive direcion) and i velociy i decreaing. If we know ha he magniude of he acceleraion i 5.6 m/ 2, hen we know ha he acceleraion i negaive (5.6 m/ 2, or 5.6 m/ 2 [E]). When you are uing vecor quaniie in equaion, you mu be careful when indicaing direcion. The following problem illurae hi. GO 376 Uni D Moion

5 SAMPLE PROBLEM 2 Deermine he Time Inerval A golf ball rolled acro he green wih an acceleraion of 1.4 m/ 2. If he iniial velociy wa 2.8 m/, for how many econd did he golf ball roll before opping? Soluion Since he acceleraion i given a 1.4 m/ 2, we know ha he direcion forward mu be poiive. Subiue he value ino he acceleraion equaion. v f v v f v a av 0m / 2. 8 m/ 1.4 m/ The golf ball rolled for 2.0 before opping. Pracice A kier ravelling a a velociy of 18 m/ op wih an acceleraion of 4.9 m/ 2. Over wha ime inerval did he kier op? A you can ee, if he acceleraion in Sample Problem 2 had no been negaive, he change in ime would have been calculaed o be a negaive number, which would mean ha ime wen backward! GO If you would like o learn more abou negaive acceleraion, go o GO Solving Problem for Velociy There are wo differen way of wriing he equaion for acceleraion: v and a v v f av We can ue eiher equaion o olve for average acceleraion. We can alo rearrange he equaion o olve problem for final velociy a hown: v v f v v f v Thi equaion i much more convenien for finding he final velociy wihou doing he algebra every ime! 13.1 Acceleraing Objec 377

6 STUDY TIP Once you have compleed a ample problem, ry hi handy rick for remembering i for an exam: On one ide of a udy card, wrie he caegory of he problem (e.g., finding average acceleraion). On he oher ide, wrie ou he problem and olve i. SAMPLE PROBLEM 3 Deermine he Final Velociy A baeball i hrown upward wih an iniial velociy of 15 m/. Wha will he baeball velociy be afer 2.0? The acceleraion of graviy i 9.8 m/ 2 down. Soluion Le upward be poiive. Subiue he value ino he final velociy equaion. v f v 15 m/ (9.8 m/ ) v f 4.6 m/ The velociy will be 4.6 m/, or 4.6 m/ downward. Pracice A lingho hoo a marble upward wih an iniial velociy of 20 m/. Wha i he peed of he marble afer 1.5? The acceleraion of graviy i 9.8 m/ 2 down. SAMPLE PROBLEM 4 Deermine he Iniial Velociy Afer acceleraing a 4.5 m/ 2 [E] for 1.4, a car velociy wa 15 m/ [W]. Wha wa he iniial velociy of he car? Soluion Le he direcion we be poiive. Subiue he value ino he final velociy equaion. v f v v v f 15 m/ (4.5 m/ ) v 21 m/ The car iniial velociy wa 21 m/, or 21 m/ [W]. Pracice A rocke wa ravelling upward when he econd age fired. The econd age gave he rocke an average acceleraion of 19 m/ 2 for 4.5. If he final velociy of he rocke wa 125 m/, wha wa he rocke velociy before he econd age fired? v i Velociy of a Verical Projecile Figure 5 Velociy ime graph for a verical projecile Graphing he Velociy of Verical Projecile If an objec i hrown upward wih an iniial velociy (v i ), he objec will rie upward and i velociy will decreae unil he objec op a ome ime,. Thi i hown in Figure 5. Afer he objec op briefly, i begin o fall. The objec will peed up a i fall, bu a you can ee in he graph, he velociy of he objec coninue o decreae. Therefore, he acceleraion i negaive and conan. Noe ha he lope of he line i negaive. For objec on Earh, he acceleraion of graviy i 9.8 m/ 2 (he negaive ign indicae ha he acceleraion i down). 378 Uni D Moion

7 13.1 CHECK YOUR Underanding 1. Wrie a definiion of acceleraion in your own word. 2. Two uden ge in an elevaor on he ground floor and ride he elevaor up o he fifh floor. (a) Decribe wha happen o heir velociy and acceleraion during he rip. (b) Skech a velociy ime graph for he rip. Wha do you hink an acceleraion ime graph for he rip would look like? Skech your idea. 3. Which of he following are poible uni for acceleraion? A. cm / B. m C. km/h/ min D. m in/ km 4. Wha happen o he velociy of an objec a i fall if he only force acing on he objec i graviy? Wha happen o he acceleraion of he objec? 5. A cycli ar from re and reache a velociy of 18 m/ [SW] in 3.8. Wha wa he cycli acceleraion? 6. A UFO i flying a a velociy of 45 m/ [E]. If 5.9 laer i velociy wa 35 m/ [W], wha wa i acceleraion? Wha aumpion did you make? 7. A moorcycle and rider ar from re and reach a velociy of 52 km/h [E] in 2.7. Wha wa he acceleraion of he moorcycle in m/ 2? 8. A priner finihe a race wih a velociy of +8.9 m/. The priner acceleraed o a op a a rae of 2.7 m/ 2. How long did i ake he priner o come o a op? 9. A mall rock wa dropped and i poiion recorded in ime inerval of 0.1. The daa i hown in Table 3. Table 3 Moion of a Rock Poiion (cm down) Average velociy (cm/ down) Average acceleraion (cm/ 2 down) (a) Draw a poiion ime graph. (b) Wha happen o he lope of he line a ime increae? (c) Wha happen o he velociy of he rock a ime increae? (d) Deermine he average velociy for each ime inerval. (e) Why i here one le average velociy value han he number of ime or poiion value? (f) Draw a velociy ime graph for your value. Draw a line of be fi for your daa. (g) Wha i he lope of he line for he velociy ime graph? (h) Why do you hink ha he iniial velociy did no equal o 0 cm/? (i) Deermine he average acceleraion for each ime inerval Acceleraing Objec 379