Kinematics in One Dimension

Transcription

1 Kinemaics in One Dimension PHY 7 - d-kinemaics - J. Hedberg - 7. Inroducion. Differen Types of Moion We'll look a:. Dimensionaliy in physics 3. One dimensional kinemaics 4. Paricle model. Displacemen Vecor. Displacemen in -D. Disance Traveled 3. Speed and Velociy....wih a direcion 4. Change in velociy.. Acceleraion. Acceleraion, he mah. 3. Slowing down 4. Acceleraion in he negaive 5. Summary of acceleraion signage. 5. Kinemaic equaions. Equaions of Moion (-D) 6. Solving Problems 7. Ploing 8. Free Fall. Drop a wrench. How high was his? 3. Every poin on a line has a angen Inroducion Moion: change in posiion or orienaion wih respec o ime. Vecors have given us some basic ideas abou how o describe he posiion of objecs in he universe/ Now, we'll coninue by exending hose ideas o accoun for changes in ha posiion. Of course he world would be awfully boring if he posiion of everyhing was consan. Differen Types of Moion We'll look a: Linear Circular Projecile Roaional Linear moion involves he change in posiion of an objec in one direcion only. An example would be a rain on a sraigh secion of he rack. The change in posiion is only in he horizonal direcion. Projecile moion occurs when objecs are launched in he graviaional field near he earhs surface. They experience moion in boh he horizonal and he verical direcions. Circular moion occurs in a few specific cases when an objec ravels in a perfec circle. Some special mah can be used in hese cases. Roaional moion implies ha he body in quesion is roaing around an axis. The axis doesn' necessary need o pass hrough he objec.... or a combinaion of hem. Page

2 Dimensionaliy in physics PHY 7 - d-kinemaics - J. Hedberg - 7 Prelude o advanced physics and engineering: Laer on, you'll have o expand your noion of dimensions a bi. I won' simply mean sraigh or curvy, bu will insead be used o describe he degrees of freedom in a sysem. For example, an orbiing body, hough i moves in a circle which requires x and y values o describe, can also be described by considering he radius and he angle of roaion insead. This is jus anoher coordinae sysem: polar coordinaes (usually: r and θ). If we describe he orbiing plane in his sysem, and say, i's going around in a perfec circle, hen he r value doesn' change and he θ value become he only dimension of ineres. Le's hold off on his approach for now, bu when i comes back laer on, welcome i wih open arms because i allows for much more powerful and simple analysis of sysems. One dimensional kinemaics For he case of -dimensional moion, we'll only consider a change of posiion in one direcion. I could be any of he hree coordinae axes. Jus a descripion of he moion, wihou aemping o analyze he cause. To describe moion we need:. Coordinae Sysem (origin, orienaion, scale). he objec which is moving z y x x (m) +x (m) d kinemaics will be our saring poin. I is he mos sraighforward and easies mahemaically o deal wih since only one posiion variable will be changing wih respec o ime. Page

3 Paricle model PHY 7 - d-kinemaics - J. Hedberg - 7 We'll need o use an absracion: All real world objecs ake up space. We'll assume ha hey don'. In oher words, hings like cars, cas, and ducks are jus poin-like paricles. This is our firs real absracion. Again, since we are rying o predic everyhing, we would like o figure ou he rules ha describe how any objec would move. Take a rain for example. If we asked a quesion like "when does he C rain ener 59h sree saion?", a naural follow-up would be "well, do you mean he fron of he rain, or he middle of he rain, or he end of he rain? Each of hese answers migh be differen by a few seconds. How do we deal wih his? By considering he rain o be a 'poin', we can neglec he acual lengh of he rain and focus on wha's more inersing: how he rain moves. The goal is o find he underlying physics ha describes all rains. Once we do ha, hen we can improve our model by including informaion abou he lengh of he individual rain we are ineresed in. Displacemen Vecor To quanify he moion, we'll sar by defining he displacemen vecor. x x Δx = x x In he case of our wandering bug, his would be he difference beween he final posiion and he iniial posiion. This figure shows he displacemen vecor Δx. This migh be differen han he disance raveled by he bug (shown in he doed line). Noe on Noaion! x f is he same hing as x x i is he same hing as x When describing moions, we usually have an iniial posiion and a final posiion. We can call hese respecively, when we do our algebra. Or anoher way of wriing hese quaniies is o say our iniial posiion is a slighly more general way of wriing hings. x x i and x f and our final posiion is jus x. This is Page 3

4 Displacemen in -D PHY 7 - d-kinemaics - J. Hedberg x (m) +x (m) Here's a car ha moves from x Δx = x The car hen reverses o x =. o x creaing a displacemen vecor of: x = 6 m m = 6 m x (m) +x (m) The leads o a displacemen vecor of Δx = 8m. Abou noaion. Δx ("dela x") refers o he change in x. Tha is, difference beween a final and iniial value: Δx = x x Or, in words, he final x posiion minus he original x posiion is equal o he change in x. Disance Traveled To ge he disance raveled, we jus nee o ake he magniude of he displacemen during a cerain moion. Δx = Disance Traveled This equaion will only be rue if he displacemen is always in he same direcion. If however, he displacemen vecor were o change direcion during a rip, he he disance raveled migh no be equal o he oal displacemen. For example, if you walk fee forward, hen urn around and walk 5 backwards. You displacemen from he iniial o final posiion will only be 5 fee, bu you will have walked a oal of 5 fee. Speed and Velociy Disance in a given ime Average Speed Elapsed ime The 'elapsed ime' is deermined in he same way as he disance: Δ =. Again, is he saring ime, and is he final ime. Page 4

5 Example Problem #: PHY 7 - d-kinemaics - J. Hedberg - 7 Taking he A rain beween 59h and 5h akes abou 8 minues. The C, which is a local, akes minues (on a good day). Find he average speed for boh of hese rips....wih a direcion Calculaing he average speed didn' ell us anyhing abou he direcion of ravel. For his, we'll need average velociy. Average Velociy Displacemen Elapsed ime In mahemaical erms: x x v Δx = Δ (SI unis of average velociy are m/s) In one-dimension, velociy can eiher be in he posiive or negaive direcion. Quick Quesion x(m) 3 This is a graph showing he posiion of an objec wih respec o ime. Which choice bes describes his moion? a) The objec is moving a.5 m/s in he +x direcion. b) The objec is moving a. m/s in he +x direcion. c) The objec is moving a. m/s in he +x direcion. d) The objec is no moving a all. 4 6 (s) Quick Quesion x(m) 3 This is a graph showing he posiion of an objec wih respec o ime. Which choice bes describes his moion? a) The objec is moving a.5 m/s in he +x direcion. b) The objec is moving a. m/s in he +x direcion. c) The objec is moving a. m/s in he +x direcion. d) The objec is no moving a all. 4 6 (s) Page 5

6 PHY 7 - d-kinemaics - J. Hedberg - 7 Thinking abou he A rain, i's clear ha is speed and velociy sayed essenially consan beween 59h and 5h ideally). However, he C rain had o sar and sop a 7 saions. To quanify, his difference in moion, we'll need o inroduce he concep of insananeous velociy. If we imagine making many measuremens of he velociy over he course of he ravel, by reducing he Δx we are considering, hen we can begin o see how we can more accuraely assess he moion of he rain. The concep of insananeous velociy involves considering an infiniesimally small secion of he moion: Δx dx v = lim = Δ Δ d This will enable us o alk abou he velociy a a paricle's posiion raher han for an enire rip. In general, his is wha we'll mean when we say 'velociy' or 'speed'. Quick Quesion 3 x(m) (s) A which of he following imes is he speed of his objec he greaes? a) = b) = s c) = 4 s d) = 6 s e) = 8 s Page 6

7 Change in velociy. PHY 7 - d-kinemaics - J. Hedberg - 7 Naurally, in order o begin moving, an objec mus change is velociy. Here's a graph of a bicyclis riding a a consan velociy. (In his case i's m/s) s s s 3 4 +x (m) Now, here's a graph of he same bicyclis riding and changing his velociy during he moion s s s 3s 3 4 +x (m) In he upper moion graph, noice how he lengh of he displacemen vecor d is he same a each inerval in ime. Meaning, ha afer second has passed, he displacemen is m, afer anoher second passes, anoher meers displacemen has occurred, making he oal displacemen equal o m. This is moion a a consan velociy. This also apparen in he lengh of he velociy vecors a each poin. They are always he same. In he boom graph, he displacemen, and velociy vecors, change each ime hey are measured. This is represenaive of moion wih non-consan velociy. The velociy is changing as ime moves on. Acceleraion This change in velociy we'll call acceleraion, and we can define i in a very similar way o our definiion of velociy: v v a = Δv = Δ Again, in his case we're alking abou average acceleraion. Example Problem #: A =, he A rain is a res a 59h sree. 5 seconds laer, i's raveling norh a 9 meers per second. Wha is he average acceleraion during his ime inerval? If we considered he same very small change in ime, he infiniesimal change, hen we could alk abou insananeous acceleraion Δv dv a = lim = Δ Δ d The SI unis of acceleraion are meers per second per second, or. Tha's probably a lile bi of a weird uni, bu, i makes sense o hink abou like his: ms m ( s ) s or vel s Page 7

8 velociy (m/s) Quick Quesion ime (s) 3 This is a graph showing he velociy of an objec wih respec o ime. Which choice bes describes his moion? a) The objec is moving a he same velociy, which is 3 m/s. b) The objec sars a res, and increases is velociy, for ever. c) The objec sars a res, hen increases is velociy for a while, hen sops moving afer 3 seconds. d) The objec sars a res, hen increases is velociy, hen moves a he same speed afer = 3s. e) The objec is no moving a all. PHY 7 - d-kinemaics - J. Hedberg - 7 Acceleraion, he mah. To quanify o he acceleraion of a moving body, say his car, we'll need o know is iniial and final velociies Play/Pause The car has a build in speedomeer, so we can look a ha o ge he speed, and if we don' change direcion, hen he velociy will be always poined in he same direcion. For his case of a car saring from res, and hen increasing velociy, he acceleraion will be a posiive quaniy. v v a = mph mph mph = = s s s 9m/s a = = +4.5m s Slowing down Wha if we ask abou a car slowing down. Now, our v =. Play/Pause Now he mah looks like his: We noice ha he acceleraion is negaive. s v = +9m/s while v v a = m/s 9m/s 9m/s = = = 4.5m/ s s s s Le's graphically subrac he velociy vecors: Now we'll subrac hem for he car slowing down. Page 8

9 Acceleraion in he negaive PHY 7 - d-kinemaics - J. Hedberg - 7 Wha if he car sars acceleraing in he negaive direcion? Now, even he speed is increasing, he velociy is geing more negaive. If we do he mah, we'll see ha he acceleraion vecor poins in he negaive direcion. Summary of acceleraion signage. When he signs of an objec s velociy and acceleraion are he same (in same direcion), he objec is speeding up When he signs of an objec s velociy and acceleraion are opposie (in opposie direcions), he objec is slowing down and speed decreases +v x +v x +v x +v x Quick Quesion 5 A one paricular momen, a subway rain is moving wih a posiive velociy and negaive acceleraion. Which of he following phrases bes describes he moion of his rain? Assume he fron of he rain is poining in he posiive x direcion. a) The rain is moving forward as i slows down. b) The rain is moving in reverse as i slows down. c) The rain is moving faser as i moves forward. d) The rain is moving faser as i moves in reverse. e) There is no way o deermine wheher he rain is moving forward or in reverse. Quick Quesion 6 A one paricular momen, a subway rain is moving wih a negaive velociy and posiive acceleraion. Which of he following phrases bes describes he moion of his rain? Assume he fron of he rain is poining in he posiive x direcion. a) The rain is moving forward as i slows down. b) The rain is moving in reverse as i slows down. c) The rain is moving faser as i moves forward. d) The rain is moving faser as i moves in reverse. e) There is no way o deermine wheher he rain is moving forward or in reverse. Quick Quesion 7 A car is moving in he negaive direcion bu slowing down. Which way is he acceleraion vecor direced? a) Posiive b) Negaive c) Acceleraion is equal o. Page 9

10 PHY 7 - d-kinemaics - J. Hedberg - 7 Quick Quesion 8 velociy (m/s) 8 4 Wha is he average velociy of his objec beween and 3 seconds? a) m/s b) 3 m/s c) 4 m/s d) 6 m/s e) m/s ime (s) 3 Kinemaic equaions v v. a = a = v = v + a x x. v = x x = v = ( + v) v We can do a lo by rearranging hese equaions. Puing v from () ino () will give us: 3. x x = v + a or, solving () for, hen insering ha ino () will give us: 4. v = v + a(x x ) v = v + a x = v = ( + v) v x = x + v + a v = v + ax v = v + a Here we have an equaion for velociy which is changing due o an acceleraion, a. I ells us how fas somehing will be going (and he direcion) if has been acceleraed for a ime,. I can deermine an objec s velociy a any ime when we know is iniial velociy and is acceleraion Does no require or give any informaion abou posiion Ex: How fas was he car going afer seconds while acceleraing from res a m/s Ex: How long did i ake o reach miles per hour Page

11 PHY 7 - d-kinemaics - J. Hedberg - 7 x = v = (v + v ) This equaion will ell us he posiion of an objec based on he iniial and final velociies, and he ime elapsed. I does no require knowing, nor will i give you, he acceleraion of he objec. Ex: How far did he duck walk if i ook seconds o reach 5 miles per hour under consan acceleraion. a x = x + v + Gives posiion a ime in erms of iniial velociy and acceleraion Doesn require or give final velociy. Ex: How far up did he rocke go? = + ax v v Gives velociy a ime in erms of acceleraion and posiion Does no require or give any informaion abou he ime. Ex: How fas was penny going when i reached he boom of he well? Equaions of Moion (-D) Things o be aware of:. They are only for siuaions where he acceleraion is consan.. The way we have wrien hem is really jus for -D moion. Equaion Missing Variable Good for finding v = v + a x a,,v x = (v + v ) a x,,v a x = x + v + v x,a, = + ax v v a,x,v Page

12 Solving Problems PHY 7 - d-kinemaics - J. Hedberg - 7. Diagram: draw a picure. Characers: Consider he problem a sory. Who are he characers? 3. Find: clearly lis symbolically wha we're looking for. 4. Solve: sae he basic idea behind soluion, in a few words (physical principles used, ec.) 5. Assess: does answer make sense? Example Problem #3: A axi is siing a a red ligh. The ligh urns green and he axi acceleraes a.5 m/s for 3 seconds. How far does i ravel during his ime? Example Problem #4: A paricle is a res. Wha acceleraion value should we give i so ha i will be meers away from is saring posiion afer.4 seconds? Example Problem #5: A subway rain acceleraes saring a x = m uniformly unil i reaches x = 35 m, a a uniform acceleraion value of.5 m/s. a. If i had an iniial velociy of m/s, wha will he duraion of his acceleraion be? b. If i had an iniial velociy of 8 m/s, wha will he duraion of his acceleraion be? Example Problem #6: If x() = , find v() and a(). Also, find he ime when he velociy is zero. Example Problem #7: A hummingbird jus noiced a brigh red flower. She acceleraes in a sraigh line owards he flower, from. m/s o 8.5 m/s a a rae of 3. m/s. How far does she ravel o reach he final velociy? Page

13 Example Problem #8: PHY 7 - d-kinemaics - J. Hedberg - 7 Traian Vuia, a Romanian Invenor, waned o reach 7 m/s in order o ake off in his flying machine. His plane could accelerae a m/s. The only runway he had access o was 8 meers long. Will he reach he necessary speed? Ploing s s s 3s 4s 5s 6s 7s 8s 9s x (f) Le's look a he moion of a honey badger. Afer each second, we noe where he honey badger is along he x axis. [s] x[f] disance [f] ime [s] Quick Quesion 9 +x Here is he posiion plo for a car in raffic. Which of he following would be he corresponding velociy graph? +v +v +v +v a b c d Quick Quesion Which of he following velociy vs. ime graphs represens an objec wih a negaive consan acceleraion? v A v B v C v D (s) (s) (s) (s) Page 3

14 PHY 7 - d-kinemaics - J. Hedberg - 7 Derive kinemaics using calculus. We can derive nearly all of kinemaics (for casses wih consan acceleraion) by considering he relaionships beween derivaives and inegrals. Le's begin wih he definiion of acceleraion: If we make he Δv and Δv infiniesimally small, dv and d, hen we can rewrie his as: a = dv d Now, we can ake he indefinie inegral of boh sides: a = Δv Δ dv = a d dv = a d Since a is assumed o be consan, we can remove from he inegrand. Performing he indefinie inegrals: where C is he consan of inegraion. To deermine he consan C, consider he equaion when =. This is he 'iniial condiion', hus he velociy a his poin will be he iniial velociy:. We herefore obain: by considering jus he definiion of acceleraion and he concep of inegraion. We can likewise consider he definiion of insananeous velociy: A similar operaion leads o: v = a + C v = v + a v = dx d v dx = v d Now, we canno remove v from his inegrand since i is no a consan value. However, we jus figured ou a relaion beween velociy and ime above, so: In his case, v dx = ( + a) d and a are boh consans. So he indefinie inegral can be solved: Again, we have a consan of inegraion o solve for: C. Le's again consider =, i.e. he iniial condiion. When =, he objec will be locaed a he iniial x posiion, x. Thus C = x. Finally, we have an equaion for x as a funcion of ime given all he iniial condiions of posiion and velociy: This is our fundamenal quadraic equaion ha describes he moion of a paricle undergoing ranslaion wih consan acceleraion. v x = v + a + C x = x + v + a Page 4

15 PHY 7 - d-kinemaics - J. Hedberg - 7 v() x() x() a v = v + a x = v x = v + velociy as a funcion of ime: v() Acceleraion is consan posiion as a funcion of ime x(). (vel. consan, accel = ) posiion as a funcion of ime Example Problem #9: A urle and a rabbi are o have a race. The urle s average speed is.9 m/s. The rabbi s average speed is 9 m/s. The disance from he saring line o he finish line is 5 m. The rabbi decides o le he urle run before he sars running o give he urle a head sar. Wha, approximaely, is he maximum ime he rabbi can wai before saring o run and sill win he race? Example Problem #: A car and a moorcycle are a x = a =. The car moves a a consan velociy v. The moorcycle sars a res and acceleraes wih consan acceleraion a. a. Find he where hey mee. b. Find he posiion x where hey mee. c. Find he velociy of he moorcycle when hey mee. This problem is asking us o describe he kinemaics of he siuaion in he mos general erms possible. There are no numbers given, so we mus do everyhing using symbolic algebra. Firs, le's make sure we undersand he seup. There are wo vehicles: a car and a moorcycle. They can be considered paricles meaning hey are poin like. The acion sars a =. A his ime, boh vehicles are locaed a he origin. The moorcycle is saionary, bu he car has a velociy, v. (* v is jus a symbol ha could be a number, like m/s or 34.3 mph. Bu we leave i as a symbol so ha we can solve his problem in a general way, applicable o any car!) Now he car will move farher han he moorcycle a firs. However, he moorcycle will cach up and overake he car because i is acceleraing. a) Find ou when, i.e. a wha ime, hey are a he same posiion. So, we need funcions ha ell us where each vehicle is locaed a a given ime. We can sar wih he basic kinemaic equaion of moion: x = x + v + a For he car, since here is no acceleraion, a =, and x =, his equaion simplifies o: x car = v For he moorcycle, i has no iniial velociy, origin: v =, bu i does has an acceleraion a. I also sars from he moo = a Page 5

16 The quesion ask when he objecs mee? Tha is, when are he x values he same. So, we can jus se he wo equaions equal o each oher. and solve his for. x moo = a. Now we have an equaion for ha we can use given any acceleraion and iniial velociy. x car b) Where does his occur? We can use he ime expression in one of he previous posiion equaions. = x moo v = a = v a PHY 7 - d-kinemaics - J. Hedberg - 7 v v x car = v = a a I should also be he same if we pu in he ime in he moorcycle's posiion equaion: v x moo = a = a = ( v a ) a c) Wha is he speed of he moorcycle? We firs need o find an equaion for speed of he moorcycles. Le's he relaionship beween posiion and velociy: So, when ime is = v a v = dx d, he speed of he moorcycle will be: Noice how he acceleraion erm is gone. The speed of he moorcycle when he wo objec mee is independen of is acceleraion. Tha's an ineresing bi of informaion ha would have been los if we did his problem using numbers insead of leers. = a v v = a = a( ) = a v x moorcycle This plo shows graphically he siuaion. We can compare he slopes ha he inersecion and see ha he slope of he moorcycle is roughly wice ha of he car. meeing ime car (s) Page 6

17 PHY 7 - d-kinemaics - J. Hedberg - 7 Quick Quesion Below is he graph of an objec moving along he x axis During which secion(s) does he objec have a consan velociy? Quick Quesion During which secion(s) is he objec speeding up? Quick Quesion 3 During which secion(s) is he objec sanding sill? Quick Quesion 4 During which secion(s) is he objec moving o he lef? (assume lef is negaive x direcion.) Free Fall A freely falling objec is any objec moving freely under he influence of graviy alone. Objec could be:. Dropped = released from res. Thrown downward 3. Thrown upward I does no depend upon he iniial moion of he objec.. The acceleraion of an objec in free fall is direced downward (negaive direcion), regardless of he iniial moion.. The magniude of free fall acceleraion is 9.8m/ = g. 3. We can neglec air resisance. 4. We'll choose our y axis o be posiive upward. 5. Consider moion near Earh s surface for now. s Page 7

18 PHY 7 - d-kinemaics - J. Hedberg - 7 Kinemaic equaion in he case of free fall: v = v g y = v = ( + v) v y = y + v g v = v gy They are he same. We jus replaced x y and a g. Quick Quesion 5 B A C D E An arrow is launched verically upward. I moves sraigh up o a maximum heigh, hen falls o he ground. The rajecory of he arrow is shown. A which poin of he rajecory is he arrow s acceleraion he greaes? Ignore air resisance; he only force acing is graviy. a) poin A b) poin B c) poin C d) poin D e) poin E f) None of hese because i is he same everywhere. Example Problem #: An objec is hrown upward a m/s: a. How long will i ake o reach he op b. How high is he op? c. How long o reach he boom? d. How fas will i be going when i reaches he boom? Quick Quesion 6 An arrow is launched verically upward. I moves sraigh up o a maximum heigh, hen falls o he ground. Which graph bes represens he verical velociy of he arrow as a funcion of ime? Ignore air resisance; he arrow is in free fall!. +v +v +v +v +v A B C D E Example Problem #: If an objec is hrown upward from a heigh y wih a speed v, when will i hi he ground? Page 8

19 Example Problem #3: PHY 7 - d-kinemaics - J. Hedberg - 7 Drop a wrench A worker drops a wrench down he elevaor shaf of a all building. a. Where is he wrench.5 seconds laer? b. How fas is he wrench falling a ha ime? Example Problem #4: A rock is hrown upward wih a velociy of 49 m/s from a poin 5 m above he ground. a. When does he rock reach is maximum heigh? b. Wha is he maximum heigh reached? c. When does he rock hi he ground? Example Problem #5: Draw posiion, velociy, and acceleraion graphs as a funcions of ime, for an objec ha is le go from res off he side of a cliff. Page 9