Chapter 2 Kinematics in One Dimension


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1 Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how fas ha how fas changes (acceleraion). We say ha an objec moing in a sraigh line is moing in one dimension, and an objec which is moing in a cured pah (like a projecile) is moing in wo dimensions. We relae all hese quaniies wih a se of equaions called he kinemaic equaions. The conen conained in all secions of chaper of he exbook is included on he AP Physics B exam. QUICK REFERENCE Imporan Terms acceleraion he rae of change in elociy acceleraion due o graiy he acceleraion of a freely falling objec in he absence of air resisance, which near he earh s surface is approximaely 1 m/s. acceleraionime graph plo of he acceleraion of an objec as a funcion of ime aerage acceleraion he acceleraion of an objec measured oer a ime ineral aerage elociy he elociy of an objec measured oer a ime ineral; he displacemen of an objec diided by he change in ime during he moion consan (or uniform) acceleraion acceleraion which does no change during a ime ineral consan (or uniform) elociy elociy which does no change during a ime ineral displacemen change in posiion in a paricular direcion (ecor) disance he lengh moed beween wo poins (scalar) free fall 19
2 Chaper Kinemaics in One Dimension moion under he influence of graiy iniial elociy he elociy a which an objec sars a he beginning of a ime ineral insananeous he alue of a quaniy a a paricular insan of ime, such as insananeous posiion, elociy, or acceleraion kinemaics he sudy of how moion occurs, including disance, displacemen, speed, elociy, acceleraion, and ime. posiionime graph he graph of he moion of an objec ha shows how is posiion aries wih ime speed he raio of disance o ime elociy raio of he displacemen of an objec o a ime ineral elociyime graph plo of he elociy of an objec as a funcion of ime, he slope of which is acceleraion, and he area under which is displacemen
3 Chaper Kinemaics in One Dimension Equaions and Symbols a x x o 1 ( o o o a o 1 a a ) x where Δx = displacemen (final posiion iniial posiion) = elociy or speed a any ime o = iniial elociy or speed = ime a = acceleraion DISCUSSION OF SELECTED SECTIONS Displacemen Disance d can be defined as oal lengh moed. If you run around a circular rack, you hae coered a disance equal o he circumference of he rack. Disance is a scalar, which means i has no direcion associaed wih i. Displacemen Δx, howeer, is a ecor. Displacemen is defined as he sraighline disance beween wo poins, and is a ecor which poins from an objec s iniial posiion x o oward is final posiion x f. In our preious example, if you run around a circular rack and end up a he same place you sared, your displacemen is zero, since here is no disance beween your saring poin and your ending poin. Displacemen is ofen wrien in is scalar form as simply Δx or x. Speed and Velociy Aerage speed is defined as he amoun of disance a moing objec coers diided by he amoun of ime i akes o coer ha disance: aerage speed = disance elapsed ime d where sands for speed, d is for disance, and is ime. 1
4 Chaper Kinemaics in One Dimension Aerage elociy is defined a lile differenly han aerage speed. While aerage speed is he oal change in disance diided by he oal change in ime, aerage elociy is he displacemen diided by he change in ime. Since elociy is a ecor, we mus define i in erms of anoher ecor, displacemen. Ofenimes aerage speed and aerage elociy are inerchangeable for he purposes of he AP Physics B exam. Speed is he magniude of elociy, ha is, speed is a scalar and elociy is a ecor. For example, if you are driing wes a 5 miles per hour, we say ha your speed is 5 mph, and your elociy is 5 mph wes. We will use he leer for boh speed and elociy in our calculaions, and will ake he direcion of elociy ino accoun when necessary. Acceleraion Acceleraion ells us how fas elociy is changing. For example, if you sar from res on he goal line of a fooball field, and begin walking up o a speed of 1 m/s for he firs second, hen up o m/s, for he second second, hen up o 3 m/s for he hird second, you are speeding up wih an aerage acceleraion of 1 m/s for each second you are walking. We wrie a 1m / s 1s 1m / s / s m 1 s In oher words, you are changing your speed by 1 m/s for each second you walk. If you sar wih a high elociy and slow down, you are sill acceleraing, bu your acceleraion would be considered negaie, compared o he posiie acceleraion discussed aboe. Usually, he change in speed is calculaed by he final speed f minus he iniial speed o. The iniial and final speeds are called insananeous speeds, since hey each occur a a paricular insan in ime and are no aerage speeds. Applicaions of he Equaions of Kinemaics for Consan Acceleraion Kinemaics is he sudy of he relaionships beween disance and displacemen, speed and elociy, acceleraion, and ime. The kinemaic equaions are he equaions of moion which relae hese quaniies o each oher. These equaions assume ha he acceleraion of an objec is uniform, ha is, consan for he ime ineral we are ineresed in. The kinemaic equaions lised below would no work for calculaing elociies and displacemens for an objec which is acceleraing erraically. Forunaely, he AP Physics B exam generally deals wih uniform acceleraion, so he kinemaic equaions lised aboe will be ery helpful in soling problems on he es. Freely Falling Bodies An objec is in free fall if i is falling freely under he influence of graiy. Any objec, regardless of is mass, falls near he surface of he Earh wih an acceleraion of 9.8 m/s, which we will denoe wih he leer g. We will round he free fall acceleraion g o 1
5 Chaper Kinemaics in One Dimension m/s for he purpose of he AP Physics B exam. This free fall acceleraion assumes ha here is no air resisance o impede he moion of he falling objec, and his is a safe assumpion on he AP Physics B es unless you are old differenly for a paricular quesion on he exam. Since he free fall acceleraion is consan, we may use he kinemaic equaions o sole problems inoling free fall. We simply need o replace he acceleraion a wih he specific free fall acceleraion g in each equaion. Remember, anyime a elociy and acceleraion are in opposie direcions (like when a ball is rising afer being hrown upward), you mus gie one of hem a negaie sign. 3
6 Chaper Kinemaics in One Dimension Example 1 A girl is holding a ball as she seps ono a all eleaor on he ground floor of a building. The girl holds he ball a a heigh of 1 meer aboe he eleaor floor. The eleaor begins acceleraing upward from res a 3 m/s. Afer he eleaor acceleraes for 5 seconds, find (a) he speed of he eleaor (b) he heigh of he floor of he eleaor aboe he ground. A he end of 5 s, he girl les go of he ball from a heigh of 1 meer aboe he floor of he eleaor. If he eleaor coninues o accelerae upward a 3 m/s, describe he moion of he ball (c) relaie o he girl s hand, (d) relaie o he ground. (e) Deermine he ime afer he ball is released ha i will make conac wih he floor. (f) Wha is he heigh aboe he ground of he ball and floor when hey firs make conac? Soluion: (a) o a 3m / s 5s 15 m / s upward (b) y 1 1 o a 3m / s 5s 37. 5m (c) When he girl releases he ball, boh she and he ball are moing wih a speed of 15 m/s upward. Howeer, he girl coninues o accelerae upward a 3 m/s, bu he ball ceases o accelerae upward, and he ball s acceleraion is direced downward a g = 1 m/s, ha is, i is in free fall wih an iniial upward elociy of 15 m/s. Therefore he ball will appear o he girl o fall downward wih an acceleraion of 3 m/s ( 1 m/s ) = 13 m/s downward, and will quickly fall below her hand. (d) Someone waching he ball from he ground would simply see he ball rising upward wih an iniial elociy of 15 m/s, and would wach i rise o a maximum heigh, a which poin i would be insananeously a res (proided i doesn srike he floor of he eleaor before i reaches is maximum heigh). (e) When he ball is released, i is raeling upward wih a speed of 15 m/s, has a downward acceleraion of 13 m/s relaie o he floor, and is a a heigh y = 1 m aboe he floor. The ime i akes o fall o he floor is 4
7 Chaper Kinemaics in One Dimension 1 y a 1 1m 13 m / s.4 s (f) In his ime of.4 s, he eleaor floor has moed up a disance of 1 1 y ae 3m / s.4s. 4m Thus, he ball and eleaor floor collide a a heigh aboe he ground of 37.5 m +.4 m = m. Graphical Analysis of Velociy and Acceleraion Le s ake some ime o reiew how we inerpre he moion of an objec when we are gien he informaion abou i in graphical form. On he AP Physics B exam, you will need o be able o inerpre hree ypes of graphs: posiion s.ime, elociy s. ime, and acceleraion s. ime. Posiion s. ime Consider he posiion s. ime graph below: x (m) Δx x (m) P Δx Δ Δ (s) (s) x The slope of he graph on he lef is, and is herefore elociy. The cured graph on he righ indicaes ha he slope is changing. The slope of he cured graph is sill elociy, een hough he elociy is changing, indicaing he objec is acceleraing. The insananeous elociy a any poin on he graph (such as poin P) can be found by drawing a angen line a he poin and finding he slope of he angen line. 5
8 Chaper Kinemaics in One Dimension Velociy s. ime Consider he elociy s. ime graph below: (m/s) Δ (m/s) Δ (s) Area (s) As shown in he figure on he lef, he slope of a elociy s. ime graph is, and is herefore acceleraion. As shown on he figure on he righ, he area under a elociy s. m ime graph would hae unis of s m, and is herefore displacemen. s Acceleraion s. ime Since he AP Physics B exam generally deals wih consan acceleraion, any graph of acceleraion s. ime on he exam would likely be a sraigh horizonal line: a (m/s ) +5 m/s a (m/s ) (s) 5 m/s (s) This graph on he lef ells us ha he acceleraion of his objec is posiie. If he objec were acceleraing negaiely, he horizonal line would be below he ime axis, as shown in he graph on he righ. 6
9 Chaper Kinemaics in One Dimension Example Consider he posiion s. ime graph below represening he moion of a car. Assume ha all acceleraions of he car are consan. G H I J x(m) C D E F B K (s) A On he axes below, skech he elociy s. ime and acceleraion s. ime graphs for his car. (m/s) (s) a(m/s ) (s) 7
10 Chaper Kinemaics in One Dimension Soluion: The car sars ou a a disance behind our reference poin of zero, indicaed on he graph as a negaie displacemen. The elociy (slope) of he car is iniially posiie and consan from poins A o C, wih he car crossing he reference poin a B. Beween poins C and D, he car goes from a high posiie elociy (slope) o a low elociy, eenually coming o res ( = ) a poin D. A poin E he car acceleraes posiiely from res up o a posiie consan elociy from poins F o G. Then he elociy (slope) decreases from poins G o H, indicaing he car is slowing down. I is beween hese wo poins ha he car s elociy is posiie, bu is acceleraion is negaie, since he car s elociy and acceleraion are in opposie direcions. The car once again comes o res a poin H, and hen begins gaining a negaie elociy (moing backward) from res a poin I, increasing is speed negaiely o a consan negaie elociy beween poins J and K. A K, he car has reurned o is original saring posiion. The elociy s. ime graph for his car would look like his: (m/s) B C F G A D E H I (s) J K The acceleraion s. ime graph for his car would look like his: a(m/s ) E F A B C D G H I J K (s) 8
11 Chaper Kinemaics in One Dimension CHAPTER REVIEW QUESTIONS For each of he muliple choice quesions below, choose he bes answer. Unless oherwise noed, use g = 1 m/s and neglec air resisance. 1. Which of he following saemens is rue? (A) Displacemen is a scalar and disance is a ecor. (B) Displacemen is a ecor and disance is a scalar. (C) Boh displacemen and disance are ecors. (D) Neiher displacemen nor disance are ecors. (E) Displacemen and disance are always equal.. Which of he following is he bes saemen for a elociy? (A) 6 miles per hour (B) 3 meers per second (C) 3 km a 45 norh of eas (D) 4 km/hr (E) 5 km/hr souhwes 3. A jogger runs 4 km in.4 hr, hen 8 km in.8 hr. Wha is he aerage speed of he jogger? (A) 1 km/hr (B) 3 km/hr (C) 1 km/hr (D).1 km/hr (E) 1 km/hr 5. A bus saring from a speed of +4 m/s slows o 6 m/s in a ime of 3 s. The aerage acceleraion of he bus is (A) m/s (B) 4 m/s (C) 6 m/s (D) m/s (E) 6 m/s 6. A rain acceleraes from res wih an acceleraion of 4 m/s for a ime of s. Wha is he rain s speed a he end of s? (A).5 m/s (B) 4 m/s (C).5 m/s (D).8 m/s (E) 8 m/s 7. A fooball player sars from res 1 meers from he goal line and acceleraes away from he goal line a 5 m/s. How far from he goal line is he player afer 4 s? (A) 6 m (B) 3 m (C) 4 m (D) 5 m (E) 6 m 4. A moorcycle sars from res and acceleraes o a speed of m/s in a ime of 8 s. Wha is he moorcycle s aerage acceleraion? (A) 16 m/s (B) 8 m/s (C) 8 m/s (D).5 m/s (E).4 m/s 9
12 Chaper Kinemaics in One Dimension 8. A ball is dropped from res. Wha is he acceleraion of he ball immediaely afer i is dropped? (A) zero (B) 5 m/s (C) 1 m/s (D) m/s (E) 3 m/s Quesions 9 11: A ball is hrown sraigh upward wih a speed of +1 m/s. 1. Which wo of he following pairs of graphs are equialen? (A) x 9. Wha is he ball s acceleraion jus afer i is hrown? (A) zero (B) 1 m/s upward (C) 1 m/s downward (D) 1 m/s upward (E) 1 m/s downward (B) x 1. How much ime does i ake for he ball o rise o is maximum heigh? (A) 4 s (B) 1 s (C) 1 s (D) s (E) 1. s (C) (D) x x 11. Wha is he approximae maximum heigh he ball reaches? (A) 4 m (B) 17 m (C) 1 m (D) 7 m (E) 5 m (E) x 3
13 Chaper Kinemaics in One Dimension Quesions 13 14: Consider he elociy s ime graph below: 13. A which ime(s) is he objec a res? (A) zero (B) 1 s (C) 3 s o 4 s (D) 4 s only (E) 8 s 14. During which ineral is he speed of he objec decreasing? (A) o 1 s (B) 1 s o 3 s (C) 3 s o 4 s (D) 4 s o 8 s (E) he speed of he objec is neer decreasing in his graph 31
14 Chaper Kinemaics in One Dimension Free Response Quesion Direcions: Show all work in working he following quesion. The quesion is worh 15 poins, and he suggesed ime for answering he quesion is abou 15 minues. The pars wihin a quesion may no hae equal weigh. 1. (15 poins) A car on a long horizonal rack can moe wih negligible fricion o he lef or o he righ. During he ime inerals when he car is acceleraing, he acceleraion is consan. The acceleraion during oher ime inerals is also consan, bu may hae a differen alue. Daa is aken on he moion of he car, and recorded in he able below. Displacemen Velociy ime x(m) (m/s) (s)
15 Chaper Kinemaics in One Dimension (a) Plo hese daa poins on he s graph below, and draw he besfi sraigh lines beween each daa poin, ha is, connec each daa poin o he one before i. The acceleraion is consan or zero during each ineral lised in he daa able. (b) Lis all of he imes beween = and = 1 s a which he car is a res. (c) i. During which ime ineral is he magniude of he acceleraion of he car he greaes? ii. Wha is he alue of his maximum acceleraion? (d) Find he displacemen of he car from x = a a ime of 1 s. (e) On he following graph, skech he acceleraion s. ime graph for he moion of his car from = o = 1 s. 33
16 Chaper Kinemaics in One Dimension ANSWERS AND EXPLANATIONS TO CHAPTER REVIEW QUESTIONS Muliple Choice 1. B Displacemen is he sraighline lengh from an origin o a final posiion and includes direcion, whereas disance is simply lengh moed.. E Velociy is a ecor and herefore direcion should be included. 3. A Aerage speed is oal disance diided by oal ime. The oal disance coered by he jogger is 1 km and he oal ime is 1. hours, so he aerage speed is 1 km/hr. 4. D a m / s 8 s m.5 s 5. E a f o 6m/ s 4m/ s 3s m 6 s 6. E f i a 4m/ s s 8m/ s 34
17 Chaper Kinemaics in One Dimension 7. D m x x 1 1 f o o a (1 m) 5 4s 5 m s 8. C The acceleraion due o graiy is 1 m/s a all poins during he ball s fall. 9. C Afer he ball is hrown, he only acceleraion i has is he acceleraion due o graiy, 1 m/s. 1. E A he ball s maximum heigh, f =. Thus, g f o 1m / s 1m / s 1. s 11. D 1 1 m y g 1 1. s 7. m 7 m s 1. B Boh of hese graphs represen moion ha begins a a high posiie elociy, and slows down o zero elociy. 13. B The line crosses he axis ( = ) a a ime of 1 second. 14. A The objec begins wih a high negaie (backward) elociy a =, hen is speed decreases o zero by a ime of 1 s. 35
18 Chaper Kinemaics in One Dimension Free Response Quesion Soluion (a) 4 poins (b) poins The car is a res when he elociy is zero, ha is, when he graph crosses he ime axis. Thus, = a 5 s, 9 s, and 1 s, as well as all poins beween 9 and 1 s. (c) i. 1 poin The acceleraion can be found by finding he slope of he s graph in a paricular ineral. The slope (acceleraion) is maximum (seepes) in he ime ineral from o 1 s. ii. poins Acceleraion = slope of s graph = m / s 1s 4m / s s m / s (d) 3 poins The displacemen of he car from x = can be found by deermining he area under he graph. Noe ha he area is negaie from o 5 s, and posiie from 5 s o 9 s. Don forge he iniial displacemen of m a =. Area from o 5 s = 1 squares =  1 m. Area from 5 o 1 s =.5 squares = +.5 m Toal displacemen from x = is m 1 m +.5 m = m. 36
19 Chaper Kinemaics in One Dimension (e) 3 poins 37
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