Physics for Scientists and Engineers. Chapter 2 Kinematics in One Dimension

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1 Physics for Scieniss and Engineers Chaper Kinemaics in One Dimension Spring, 8 Ho Jung Paik

2 Kinemaics Describes moion while ignoring he agens (forces) ha caused he moion For now, will consider moion in one dimension Along a sraigh line Will use he paricle model A paricle is a poin-like objec, has mass bu infiniesimal size 7-Jan-8 Paik p.

3 Acceleraion and Velociy 7-Jan-8 Paik p. 3

4 Moion Diagrams Posiion graphs 7-Jan-8 Paik p. 4

5 Inerpreing a Posiion Graph 7-Jan-8 Paik p. 5

6 Posiie & Negaie Slopes 7-Jan-8 Paik p. 6

7 Uniform & Nonuniform Moion 7-Jan-8 Paik p. 7

8 Uniform Moion A moion wih f Δ f Δ Δ Δ i r consan Consider -D moion in direcion i 7-Jan-8 Paik p. 8

9 Collision Problem Bob leaes home in Chicago a 9: am and raels eas a a seady 6 mph. Susan, 4 miles o he eas in Pisburgh, leaes a he same ime and raels wes a a seady 4 mph. Where will hey mee for lunch? 7-Jan-8 Paik p. 9

10 7-Jan-8 Paik p. Posiion s Time and Mah 4 mi ) ( h) ( mi ) ( S S S S B B B B B 4. h 4 mph)] ( [6 mph 4 mi 4 mi ) ( 4 mi S B S B S B S B B B 4 mi 4. h 6 mph They mee a and when B S. Using his in Bob s equaion 4. h 4 mph)] ( [6 mph 4 mi 4 mi ) ( 4 mi S B S B S B

11 Insananeous Velociy The limi of he aerage elociy as he ime ineral becomes infiniesimally shor. Δ Δ lim Δ d d Slope of he green line The insananeous elociy indicaes wha is happening a eery poin of ime. 7-Jan-8 Paik p.

12 Insananeous Velociy, con The insananeous speed is he magniude of he insananeous elociy The aerage speed is no always he magniude of he aerage elociy! 7-Jan-8 Paik p.

13 Posiion from Velociy Since he insananeous elociy is d d he change in posiion of a moing objec is gien by d d d 7-Jan-8 Paik p. 3 d d

14 Moion Equaion from Calculus Displacemen equals he area under he elociy ime cure The limi of he sum is a definie inegral: 7-Jan-8 Paik p. 4

15 Insananeous Acceleraion Insananeous acceleraion is he limi of he aerage acceleraion as Δ approaches : a Since lim Δ Δ Δ d, d d d a d d 7-Jan-8 Paik p. 5

16 Insananeous Acceleraion, con The slope of he elociy s ime graph is he acceleraion The blue line is he aerage acceleraion beween i and f The green line represens he insananeous acceleraion a f 7-Jan-8 Paik p. 6

17 Acceleraion and Velociy, Equal ime delay snapshos The car is moing wih consan posiie elociy (red arrows mainaining he same size) Acceleraion equals zero 7-Jan-8 Paik p. 7

18 Acceleraion and Velociy, Equal ime delay snapshos Velociy and acceleraion are in he same direcion Acceleraion is posiie and uniform (blue arrows mainaining he same lengh) Velociy is posiie and increasing (red arrows geing longer) 7-Jan-8 Paik p. 8

19 Acceleraion and Velociy, 3 Equal ime delay snapshos Acceleraion and elociy are in opposie direcions Acceleraion is negaie and uniform (blue arrows mainaining he same lengh) Velociy is posiie and decreasing (red arrows geing shorer) 7-Jan-8 Paik p. 9

20 Consan Acceleraion For consan acceleraion, he aerage elociy can be epressed as he arihmeic mean of he iniial and final elociies ae No rue if a is no consan 7-Jan-8 Paik p.

21 Velociy from Acceleraion Since he insananeous acceleraion is he change in elociy is gien by d If a is a consan, a d d - ad 7-Jan-8 Paik p. ad a( ) a( ) Eq. ()

22 Displacemen from Acceleraion For consan acceleraion, Since and ae Therefore, ae ( ) ( ) ae a( ) [ ] a ) a( ) ( Eq. () ( ) a( ) 7-Jan-8 Paik p.

23 7-Jan-8 Paik p. 3 Wih Time Eliminaed From Eq. (), Subsiuing his ino Eq. (), Therefore, a ) ( a ( ) a a a a Eq. (3)

24 7-Jan-8 Paik p. 4 -D Kinemaic Equaions Wih consan acceleraion, You may need o use wo of he equaions o sole one problem Many imes here is more han one way o sole a problem ) ( (3) ) ( ) ( () ) ( () a a a

25 Displacemen Time Cure Slope of he cure is he elociy Cured line indicaes he elociy is changing Therefore, here is an acceleraion 7-Jan-8 Paik p. 5

26 Velociy Time Cure Slope is he acceleraion Sraigh line indicaes he elociy is changing a a consan rae Therefore, a consan acceleraion 7-Jan-8 Paik p. 6

27 Acceleraion Time Cure Slope is he rae of change in acceleraion Zero slope indicaes a consan acceleraion 7-Jan-8 Paik p. 7

28 Eample of a A car is moing a a consan elociy of 6 mph. The drier suddenly sees an animal crossing he road ahead. If he drier s reacion ime is. s, show how far he car will go before he drier pushes on he brake pedal. We know 6 mph (.447 m/s / mph) 7 m/s, a m/s, Use he equaion: reac. s ( ) a( ) The displacemen of he car before puing on he brakes: 7 m/s. s m 5.4 m 7-Jan-8 Paik p. 8

29 Eample of a A paricle sars from res and acceleraes as shown in he figure. Deermine (a) he paricle s speed a. s and. s, and (b) he disance raeled in he firs. s. (a) (b) a a a 3 Δ 5 Δ Δ Δ 3 Δ Δ m/s. m/s. ( 3.) 5. a a a 5. m/s ( Δ )... m ( ) Δ m ( Δ ). 5. ( 3.) m Jan-8 Paik p. 9

30 Freely Falling Objecs A freely falling objec is any objec moing freely under he influence of graiy alone, i.e., wih negligible air drag Objec has a consan acceleraion due o graiy Demonsraion : Free fall in acuum Demonsraion : Measure a 7-Jan-8 Paik p. 3

31 Acceleraion of Free Fall Acceleraion of an objec in free fall is downward, regardless of he iniial moion The magniude of free fall acceleraion is usually aken as g 9.8 m/s, howeer g decreases wih aliude g aries wih laiude 9.8 m/s is he aerage a he Earh s surface 7-Jan-8 Paik p. 3

32 Free Fall Eample A A, iniial elociy is upward ( m/s) and acceleraion is g ( 9.8 m/s ). A B, elociy is and acceleraion is g. A C, elociy has he same magniude as a A, bu is in he opposie direcion. The final displacemen is 5. m. (a) Find he disance from A o B. (b) Find he elociy a C. (c) Find he elociy a y E 5. m. 7-Jan-8 Paik p. 3

33 (a) (b) (c) B y B C C E E E Free Fall Eample, con A (. m/s) B ( 9.8 m/s ±. m/s (choose minus sign, as down) C a( y (. m/s) 4(m/s) a( y B (. m/s) (9.8 m/s a( y C E ) y A )( m.4 m) 98(m/s) 38(m/s) ± 37. m/s (choose he minus sign) ) ( 9.8 m/s ) ) ( 9.8 m/s 4(m/s) 8.6 m/s y y B C )( y m).4 m B 399.8(m/s) )( 5. m m) 7-Jan-8 Paik p. 33

34 Problem Soling Concepualize Think abou and undersand he siuaion Make a quick drawing of he siuaion Gaher he numerical informaion Wrie down he giens Focus on he epeced resul Wha are you asked o find? Think abou wha a reasonable answer should be, e.g., sign of quaniies, unis 7-Jan-8 Paik p. 34

35 Problem Soling Caegorize Simplify he problem Can you ignore air resisance? Model objecs as paricles Try o idenify similar problems you hae already soled 7-Jan-8 Paik p. 35

36 Problem Soling Analyze Selec he relean equaion(s) o apply Sole for he unknown ariable Subsiue appropriae numbers Calculae he resuls Include unis Round he resul o he appropriae number of significan figures 7-Jan-8 Paik p. 36

37 Problem Soling Finalize Check your resul Does i hae he correc unis? Does i agree wih your concepualized ideas? Does he answer hae he correc signs? Look a limiing siuaions o be sure he resuls are reasonable e.g., he correc limi if a? Compare he resul wih hose of similar problems 7-Jan-8 Paik p. 37

38 Eample A hare and a oroise compee in a. km race. The oroise crawls seadily a is maimum speed of. m/s oward he finish line. The hare runs a is maimum speed of 8. m/s oward he goal for.8 km and hen sops o ease he oroise. How close o he goal can he hare le he oroise approach before resuming he race? 7-Jan-8 Paik p. 38

39 Eample, con The hare sis a 8 m waiing for he oroise. To go he las m, he hare will ake Δ m/(8. m/s) 5. s In 5. s, he oroise can go Δ. m/s 5. s 5. m In order for he hare o win, he mus resar before he oroise ges wihin 5. m of he finish line. 7-Jan-8 Paik p. 39

40 Eample Jules Verne in 865 suggesed sending people o he Moon by firing a space capsule from a -m long cannon wih a final elociy of.97 km/s. Wha would hae been he acceleraion eperienced by he space raelers during launch? Compare your answer wih he free-fall acceleraion 9.8 m/s. f m/s, i (.97 a i 3 f a( y. 97 f m/s) y 6.3 (m/s) 44 m i ) ( m/s) 3 m/s, y.735 i 5 m, a( m) m/s m 7-Jan-8 Paik p. 4 y f.79 4 g

41 Eample 3 Speedy Sue, driing a 3. m/s, eners a one-lane unnel. She hen obseres a slow-moing an 55 m ahead raeling a 5. m/s. Sue applies her brakes bu can accelerae only a. m/s because he road is we. Will here be a collision? If yes, deermine how far ino he unnel and a wha ime he collision occurs. 7-Jan-8 Paik p. 4

42 3. ( Eample 3, con Sue' s posiion : ( ) Van' s posiion : ( ) S V C S V m (3. m/s) 55 m (5. m/s) or 5. ± C 3.6 s or.4 s The smaller one is he collision ime. The wreck happens a posiion : S V V C C Collision occurs if here is a soluion a a S V ) 55 m 5. m/s.4 s m C C 55 7-Jan-8 Paik p. 4 C o 5. C S ( C ) (. m/s V ( ( m/s C ) : ) )

43 Eample 4 A suden climbs a 5.-m cliff ha oerhangs a calm pool of waer. He hrows wo sones erically downward,. s apar, and obseres ha hey cause a single splash. The firs sone has an iniial speed of. m/s. (a) How long afer release of he firs sone do he wo sones hi he waer? (b) Wha iniial elociy mus he second sone hae? (c) Wha is he speed of each sone a he insan he wo hi he waer? 5. m y m s. s 7-Jan-8 Paik p. 43

44 Eample 4, con (a) (b) The second sone : (c) Firs sone : m. ± 5. m y. m (4.9) a( ) a( ) 4(4.9)( 5.) 5. m (. m/s)( (3. s. s) 5. m/s. m/s ( 9.8 m/s 5.3 m/s ( 9.8 m/s y y y ( ) s) )(3. s s) ( 9.8 m/s 3. s ( 9.8 m/s )(3. s. s) )(3. s. s) 7-Jan-8 Paik p. 44 ( a( ) ) )( s) or 3.4 s a( ) 3.4 m/s 34.9 m/s (Choose )

45 Eample 5 The Acela is he Porsche of American rains. I can carry 34 passengers a 7 mi/h. A elociy-ime graph for he Acela is shown in he figure. (a) Find he peak posiie acceleraion of he rain. (b) Find he rain s displacemen beween and s. 7-Jan-8 Paik p. 45

46 Eample 5, con (a) Peak acceleraion is gien by he slope of he seepes angen o he - cure. From he angen line shown, we find 5 D D 3 4 (s) a Δ Δ. (55 45) mi/h ( - 5) s 6 m 36 s s.98 m/s. (mi/h)/s 7-Jan-8 Paik p. 46

47 Eample 5, con (b) Area under he - cure equals he displacemen. 5 We approimae he area wih a series of riangles and recangles (s) Δ area area 5 mi/h 5 s mi/h 5 s 4 mi/h s area 3 area 4 (7 6) mi/h s 4 mi 36 s s area 5 5 mi/h 5 s 6 mi/h s 6.7 mi 7-Jan-8 Paik p. 47

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