Mechanics The study of objects, motion, force, and energy. Dynamics The description of how objects move

Transcription

1 Kinemaics in 1D-1 Mechanics is he sudy of he moion of ojecs and is causes. I is divided ino wo pars: kinemaics, which descries how ojecs move, and dynamics, which descries why ojecs move. Mechanics The sudy of ojecs, moion, force, and energy. Kinemaics Dynamics The descripion of how ojecs move Why ojecs move as hey do Kinemaics in 1D descries he moion of ojecs in one dimension such as a sraigh line. An ojec, as i moves, may spin aou an axis. We will neglec such a moion and deal wih only he ranslaional moion of an ojec. In sudying kinemaics, we generally deal wih wo differen kinds of physical quaniies: 1. scalars,. vecors. A scalar has only a magniude (ha is how much of i we have) whereas a vecor has oh a direcion and a magniude. In oher words, a vecor is a scalar quaniy ogeher wih a direcion. There are oher kinds of physical quaniies such as ensors. We will no sudy ensors in his course in deail. Some physical quaniies: Scalars: The disance eween wo poins. Speed: The disance covered in a given amoun of ime, i.e., disance covered per uni ime Mass Volume Vecors: Displacemen: Disance+Direcion Velociy: Speed+Direcion Acceleraion: Change in velociy per uni ime Examples for scalars: Force 1 The mass m of a person has only a magniude which can e measured in kg or ls: 70 kg., 176 ls., ec. Area Quesions: Deermine wheher he descripions The disance d eween wo ciies has only a magniude which can e measured in m, miles, ec. elow give scalars or vecors. 1 Yeserday I ran 10 miles. I walked miles due Eas. 3 To ge here, go Eas for 10km, and ake he firs Examples for vecors: urn on your righ, hen go Souh for km. I is he 1 Displacemen of an ojec is accomplished y moving he ojec over a disance in a specific direcion, from poin #1 o poin #. firs uilding on your righ. 4 On summer days, I ofen run 9.5 miles in an hour around he Mounain Lakes; from here, I ike o he When you are giving direcions o someone, you use vecors. You have o ell he person how far & in wha direcion o go, such as when o urn lef, righ or go sraigh. pool for 4 miles, and swim 1km in 0 minues. 5 Why are mass, disance, speed, and volume scalars? 6 Why are displacemen, velociy, acceleraion, force, and area vecors? , Mesu Bahadır Çakır

2 Kinemaics in 1D- Examples: Mass: The amoun of maer an ojec has. Mass is a scalar. Disance: A measure of how far apar wo poins are. Disance is a scalar. Displacemen: The movemen of an ojec from a poin o anoher poin. Displacemen is a vecor. Displacemen is a disance raveled in a cerain direcion. Speed: How fas somehing is going. Amoun of disance raveled in a given amoun of ime. Speed is a scalar. Velociy: How fas somehing is going in a cerain direcion. Velociy is speed of an ojec in a cerain direcion. Acceleraion: The change in he velociy of an ojec in a given amoun of ime. Since he velociy is a vecor, so is he acceleraion. You will learn some of he oher quaniies as we mee hem during he year. The speed, velociy, acceleraion of an ojec can e given in wo differen ways: 1. as an average. a a cerain momen in ime Average speed: The fracion of how far an ojec has raveled in a given amoun of ime. Insananeous speed: How fas an ojec is moving a a cerain momen in ime Average velociy: The fracion of how far an ojec has raveled in a given amoun of ime in a cerain direcion. Insananeous speed: How fas an ojec is moving a a cerain momen in ime in a cerain direcion Displacemen Speed average speed = Insananeous speed Velociy disance raveled ime elapsed v ins average velociy = displacemen ime elapsed Insananeous velociy Acceleraion x = x x 1 v = x = lim Δx = dx Δ 0 Δ d v ins Average Acceleraion a = v v 1 1 Insananeous acceleraion v = v 0 + a v = v + v 0 x = x 0 + v a ins x = x 0 + v a v = x = lim Δx = dx Δ 0 Δ d = lim Δv = dv Δ 0 Δ d KE 1 + PE 1 = KE + PE KE KE 1 = PE PE 1 1 m v 1 m v 0 = F. d = m a.( x x 0 ) v v 0 = a.( x x 0 ) You can also use hese expressions as given elow. However, you mus rememer ha he direcion of he vecor quaniies have no een aken ino accoun. Therefore, you mus keep rack of wha is in he posiive direcion and wha is in he negaive in direcion. v = v 0 + a v = v + v 0 x = x 0 + v x = x 0 + v a KE 1 + PE 1 = KE + PE KE KE 1 = PE PE 1 1 m v 1 m v 0 = F. d = m a (x x 0 ) v v 0 = a (x x 0 ) , Mesu Bahadır Çakır

3 Kinemaics in 1D-3 Example 1: A wolf walking due Norh covers 10 km in 1 hour. a. Oain he disance he wolf has raveled in 1h.. Oains he wolfʼs speed. c. Was he wolf running or walking? d. Oain he wolfʼs displacemen. e. Oain he wolfʼs velociy. Answers: a. x-xo =10km.. v = x x o = 10 km hr = m 60s =.8 m s c. A good amaeur runner can run 15 km/hr. A d. world class walker can walk 15 km/hr (his is an exremely fas walking pace. Therefore, we can assume ha he wolf is roing. x x o = 10km due Norh e. v = x x o = 10 km hr due N =.8 m s due Norh Prolem 1: A car ravels due Souh-Wes 10 miles in one and a half hours. a. Wha is he disance he car ravels in one and a half hours?. Wha is he carʼs speed? c. How does he car ravel in 1 hour? d. Wha are he chances ha he driver will ge a speeding icke? e. Wha is he displacemen of he car a he end of one and a half hours? f. Wha is he velociy of he car? Prolem : An alien eagle is raveling 100m/s due NE. Oain is a. is speed. velociy c. how far i goes in 1 hour d. is displacemen in 1 hour Example : A world class spriner changes his speed from 0m/s o 9m/s in 0.5 s. a. Oain his acceleraion.. Oain he disance he ravels during his 0.5s Answer: a. a = v v 1 = 9m s 0m s 0.5s = 18 m s Prolem 3: The iniial velociy of a spriner is running 1m/s. His speed reduces o o 9.5m/s in 4s. Wha is his acceleraion? Prolem 4: A deer running 1.5m/s sees a wolf. Wih an acceleraion of 0m/s, she increases her speed o 15m/s. How long did i ake he deer o reach his final speed?. x x 1 = v = v 1 + v = 4.5 m s 0.5s =.5m x x 1 = v a = m s (0.5s) =.5m Example 3: A world class is running 1m/s. She increases her speed y acceleraing 1.6m/s. a. Oain her velociy a he end of 0.3s.. Oain he disance raveled a he end of 0.3s. Answer: v = v 1 +a = 1 m s + 1.6m s 0.3s = 1.48 m s x x 1 = v = 1m s m s 0.3s = 0.37m x x 1 = v a = (0.3) = 0.37m , Mesu Bahadır Çakır

4 Prolem 1: I reach a speed of 10m/s from 0m/s in seconds. Wha is my acceleraion? Prolem : I run 9.5 miles in 61 minues. Wha is my average speed? Could you calculae my insananeous speed 15 minues afer I sared running using his informaion? A his speed, how long would i ake me o run a marahon? Prolem 3: Melissa sars walking 10 meers away from he school enrance due norh-wes. In 9 minues she ends up 100 meers away from he school enrance. a. How far did she walk?. Wha is her oal displacemen? c. Wha is her average velociy during his walk? Prolem 4 Tim drops an ojec wih zero iniial velociy. Wha will e is velociy a. 1 second laer c. 3 seconds laer. seconds laer d. 1 seconds laer Prolem 5 Jason drops an ojec wih 33 m/s iniial velociy. Wha will e is velociy a. 1 second laer c. 3 seconds laer. seconds laer d. 1 seconds laer Prolem 6 Joy hrows a aseall upward wih 30 m/s iniial velociy. Wha will e is velociy a he end of a. 1 s. s c. 3 s d. 4 s e. 5 s f. 6 s g. 1 s Prolem 7 While Oie is siing happily, he sees a grasshopper and sars walking owards i. When Oie ges oo close o he grasshopper, he grasshopper jumps sraigh up wih a 0 m/s velociy. a. Oain he grasshopperʼs velociy when i reaches he highes poin of is journey. When will he grasshopper reach he highes poin of is journey? c. Wha will e is acceleraion a ha poin? d. How high aove he ground will i e a ha poin? e. When will i land ack on he ground? f. Wha will e is velociy a he momen i lands ack on he ground? Kinemaics in 1D , Mesu Bahadır Çakır

5 Kinemaics in 1D-5 Free Fall: Near he surface of a plane, an ojec experiences free fall if he ne force acing on i is he force of graviy. In real life a falling ojec also experiences air fricion. Unless oherwise saed, we will assume ha air fricion acing on an ojec is negligily small. This is a valid assumpion if he ojec in quesion is very dense and has a small cross secional surface area. Since we have assumed ha he force acing on a freely-falling ojec is he force of graviy, is acceleraion is equal o g, i.e., a = 10 m. In oher words, is velociy s changes y 10m/s every second in he downward direcion. If he ojec is moving upward, is velociy decreases 10m/s every second unill i reaches he highes poin of is journey where is insananeous velociy is zero; hen i sars moving downward. If he ojec is moving downward, is velociy increases 10m/s every second in he downward direcion. If an ojec is dropped from a heigh ho aove ground wih a zero iniial velociy, i reaches a heigh h in a given amoun of h = h ime such ha o + 1 g v = v o v = 0 v = v o ha is h = h o 1 g Is insananeous velociy a ime is equal o v = g If he ojec has an iniial velociy v o, i displacemen will e affeced y an amoun v o. As a resul, h = h o + v o + 1 is insananeous velociy a ime is giveny g If vo is upward, If vo is downward, v = v o + g h = h o + v o 1 g v = v o g h = h o v o 1 g v = v o + g An ojec hrown upward will reach is heighes poin when is insananeous velociy reaches zero; v = v o g = 0 Tha is, when hp = v o g , Mesu Bahadır Çakır

6 Kinemaics in 1D-6 Prolem 1: An alien deer a Plaeau poin of he Grand Canyon decides o ake a swim in he river. He jus seps of he rim wih zero iniial velociy. Assume ha Plaeau poin is 500m aove he river. a. How long afer he deer seps of he rim, does she hi he surface of he waer?. Wih wha velociy does he deer hi he surface of he waer? c. If he deer goes 10m elow he surface of he waer efore she sops sinking, wha is her acceleraion inside he waer? d. Would a normal deer survive such an impac? Prolem : A he end of he Iner Galacic Baskeall Championship eween he Andremoda and Milky Way Galaxies, a memer of he winning eam hrows he askeall sraigh up wih a 10km/s iniial velociy. Assume ha he graviaional acceleraion of he plane where he game is played is g P = 30 km s and ha he askeall cour is 90km aove he surface of he plane. a. Wha is he velociy of he askeall a is maximum heigh?. When does he askeall reach is maximum heigh? c. How high aove he cour level does he askeall reach is maximum heigh? d. When does i ge ack o he cour level? e. Wha is is velociy when i ges ack o he cour level? f. If here is hole in he cour and he all goes righ hrough i, when does he askeall reach he surface of he plane? g. Wih wha velociy does he all hi he surface of he plane? h. Is he player a human eing or no? Why? Prolem 3: Assume ha in prolem, he player hrows he all downward wih a 10km/s iniial velociy. Answer he quesions f and g. How do your answer compare o he ones you oained in prolem. Why are he answers relaed in ha way? Circular Moion as 1D-Moion: We can rea he circular moion of an ojec as a onedimensional moion. In his case, he pah raveled y he ojec , Mesu Bahadır Çakır

7 { Kinemaics in 1D-7 If he velociy of an ojec is consan, is displacemen is given y The area of a recangle of sides a and is given y vo d = v o a Area= a If an ojec sars wih a vo iniial velociy and i reduces o 0 or i sars wih 0 iniial velociy and i increases o vo, he ojec s displacemen is given y The area of a righ riangle of high a and ase is given y vo vo a d = v o d = v o Area= a Area= a a If an ojec sars wih aniniial velociy v1 and i increases or reduces o v, he ojec s displacemen is given y The area of a of highs a1 and a and ase is given y v1 v d = v 1 + v or d = v a v1 v v 1 { v d = v 1 + v Area= a 1 + a Area= a 1 + a a a 1 a 1 a { a a 1 We can also wrie he area as { where he firs erm represen he disance covered wih a consan speed and he second erm represens he addiional disance covered due o acceleraion. If he acceleraion is posiive, his disance is added. If he acceleraion is negaive, his disance is suraced. Area= a a a 1 where he firs erm is he area of he square a he oom and he second erm is he area of he riangle on he op d = v a = v v v 1 = v v v , Mesu Bahadır Çakır