# One-Dimensional Kinematics

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1 One-Dimensional Kinemaics One dimensional kinemaics refers o moion along a sraigh line. Een hough we lie in a 3-dimension world, moion can ofen be absraced o a single dimension. We can also describe moion along a cured pah as onedimensional moion. Terms we will use: Posiion, disance, displacemen Speed, elociy (aerage and insananeous) Acceleraion (aerage and insananeous) 1

2 Galileo ( ) Wha mahemaical forces goern acceleraed moion? A wooden and sone sphere are dropped from he ower of pisa. Which one reaches he earh firs? Answer: same ime! 2

3 The inclined plane eperimen Disance raeled goes wih he square of ime: ()~ 2 Time measured wih waer! 3

4 Coordinaes & Vecors No only do we always use a quaniaie measure for disance and ime, we also inuiiely use ecors and coordinaes for describing locaions and displacemens. For eample, gie direcions from Teas A&M Commece o Plano, TX. Turn LEFT ono TX-24 S / TX-50 S. 9.8 miles. Merge ono I-30 W / US-67 S miles. Take he US-75 N ei- EXIT 47B- oward SHERMAN. 0.5 miles. Take he MAIN ST WEST / ELM ST ei on he LEFT. 0.2 miles. Each insrucion includes boh disance and direcion 4

5 Vecors To define a posiion in (2- dim) space, we define an origin and a coordinae sysem (,y), (N,E) ec. A poin P can be described as an ordered pair (,y) = (4m,3m) or a ecor of lengh 5 m poining 36.8 counerclockwise from he -ais. y 37 P We can hink of he poin P as eiher a posiion, or a displacemen: P is 4 m along he -ais plus 3 m along he y-ais 5

6 Coordinae Sysems A coordinae sysem is used o describe locaion. A coordinae sysem consiss of: a fied reference poin called he origin a se of aes a definiion of he coordinae ariables 6

7 Eample caresian (-3,5) y The arrow indicaes he posiie direcion. (4,2) The posiion of an objec is is locaion in a coordinae sysem. 7

8 Disance and displacemen Disance is he oal lengh of rael. I is always posiie. I is measured by he odomeer in your car. Displacemen is defined as he change in posiion of an objec. Δ f i f = final alue of, i = iniial alue of Change can be posiie, negaie or zero. Displacemen is a ecor (see Chaper 3) Δ (Dela)=change 8

9 =0 ΔX=30 Displacemen =1 X=0 X=30 X=45 Displacemen is a ecor and hus has direcion (sign) Displacemen is no equal o Disance =1 ΔX=-30 =0 X=0 X=30 X=45 9

10 Aerage Speed and Velociy Speed and elociy are no he same in physics! Speed is rae of change of disance: aerage speed = disance elaspsed ime Velociy is rae of change of displacemen: aerage elociy = displacemen elaspsed ime = (always posiie) f f i i (posiie, negaie or zero) elociy is a ecor (see Chaper 3) Here we are jus giing he -componen of elociy, assuming he oher componens are eiher zero or irrelean o our presen discussion SI unis of speed and elociy are m/s. 10

11 Eample Wha is he aerage speed of a person a he equaor due o he Earh s roaion? Disance raelled in one day (one roaion) equals circumference of earh = (2π)(radius) = (2π)( m) = m = km Aerage speed = ( km)/(24hour) = km/hr You don feel his! Velociy (in iself) is no imporan o dynamics! Wha is he aerage elociy of a person a he equaor due o he Earh s roaion? Zero 11

12 Aerage Velociy Displacemen Aerage elociy= Time ineral Δ = = Δ =0 =1 =2 =0 =1 =2 X=0 X=30 X=45 Which objec has he larges aerage elociy oer 2 s? Same! f f i i 12

13 Graphically (m) = Δ Δ = f f i i = slope (s) The aerage elociy is he slope of he line connecing begin and end poin in he - graph. Ignore wha happens in beween. 13

14 Posiion s. Time Plos The aerage elociy beween wo imes is he slope of he sraigh line connecing hose wo poins. aerage elociy from 0 o 3 sec is posiie aerage elociy from 2 o 3 sec is negaie PHYS

15 Vecors and Scalars... =1 ΔX=-30 =0 speed X=0 X=30 X=45 Δ Δ = f f i i = =-30 Aerage elociy:ecor Δ Δ f f i i = = =+30 Aerage speed:scalar 15

16 Insananeous Velociy Bu officer, I only droe 1 mile! How would you know I droe a 60 miles per hour... Someimes we wan o know he speed a one paricular poin in ime. 16

17 Insananeous elociy (m) = Δ Δ = f f i i (s) Wha is he Velociy a =2.0 s? Consider he aerage elociy for a ery small ime ineral wih =2.0 s in he cener! 17

18 Insananeous elociy (m) Δ Δ = lim Δ 0 Δ Δ (s) 18

19 Insananeous elociy (m) Δ lim Δ Δ = Δ 0 Δ = Δ Δ angen (s) =slope of he angen o he - cure a =2.0 s Think abou he sign! 19

20 Insananeous Velociy The elociy a one insan in ime is known as he insananeous elociy and is found by aking he aerage elociy for smaller and smaller ime inerals: lim Δ 0 Δ Δ The speedomeer indicaes insananeous elociy (Δ 1 s). On an s plo, he slope of he line angen o he cure a a poin in ime is he insananeous elociy a ha ime. 20

21 21 Quesion Which objec had he larges aerage elociy? Which objec had he larges insananeous elociy? A B B i f i f = Δ Δ = angen Δ Δ = B

22 Acceleraion Ofen, elociy is no consan, raher i changes wih ime. The rae of change of elociy is known as acceleraion. a a Δ Δ = f i This is he aerage acceleraion. Acceleraion is a ecor. The uni of acceleraion is: m/s 2 f i posiie, negaie or zero Car acceleraion is ofen described in unis miles per hour per sec Acceleraion of 0 o 60 miles per hour in 8 sec = (60mi/hr-0mi/hr)/8sec =7.5mi/(hr*s) mi mi 5280 f 12in m 1hr m 7.5 = 7.5 = 3.35 hr s hr s mi f in 3600s 2 s 22

23 Velociy s. Time Plos Graphically, acceleraion can be found from he slope of a elociy s. ime cure. For hese cures, he aerage acceleraion and he insananeous acceleraion are he same, because he acceleraion is consan. 23

24 Deceleraion Deceleraion refers o decreasing speed is no he same as negaie acceleraion occurs when elociy and acceleraion hae opposie signs Eample: A ball hrown up in he air. The elociy is upward bu he acceleraion is downward. The ball is slowing down as i moes upward. (Once he ball reaches is highes poin and sars o fall again, i is no longer deceleraing.) If up is our conenion for posiie, hen boh when he ball is rising and falling, he acceleraion is negaie (during he insan of bounce, he acceleraion is posiie). 24

25 Acceleraion A change in slope in he - graph means a change in elociy: Acceleraion 25

26 Acceleraion f i = Aerage acceleraion f i Insananeous acceleraion: a = lim Δ 0 Δ Δ a = Δ Δ angen 26

27 =0,=20 Quesion =1, =5 Wha is he aerage acceleraion beween =0 and =1 A) 15 B) -15 C) 0 D) infiniy 27

28 GALILEO! Quesion Which - diagram maches which a- diagram? 28

29 elociy (m/s) 6 Eample: Velociy s. Time Plo 4 2 B C 0-2 A 5 10 ime (s) -4 29

30 1. Wha is he elociy a ime = 3 sec? 2. Wha is he elociy a poin A? 3. When is he acceleraion posiie? 4. When is he acceleraion negaie? 5. When is he acceleraion zero? 6. When is he acceleraion consan? 7. When is here deceleraion? 8. Wha is he acceleraion a poin C? 9. Wha is he acceleraion a ime = 6 sec? 10. During wha 1 s ineral is he magniude of he aerage acceleraion greaes? 30

31 Eample be careful wih signs A car moes from a posiion of +4 m o a posiion of 1 m in 2 seconds. The iniial elociy of he car is 4 m/s and he final elociy is 1 m/s. (a) Wha is he displacemen of he car? (b) Wha is he aerage elociy of he car? (c) Wha is he aerage acceleraion of he car? Answer: (a) Δ = f i = 1 m (+4 m) = 5 m (b) a = Δ/Δ = 5 m/2 s = 2.5 m/s (c) a a Δ Δ = f f i i = 1 m s ( 4 2 s m s ) = 1.5 m/s 2 deceleraion! 31

32 32 Moion wih Consan Acceleraion If acceleraion is consan, here are four useful formulae relaing posiion, elociy, acceleraion a a ime : a a a a Δ + = + = + + = + + = + = 2 ) ( 2 ) ( = iniial elociy 0 = iniial posiion 0 = iniial ime assumed here o be a 0 s. If 0 0, replace in hese formulae wih 0 (Insead of f, i, we are using and 0 )

33 Noe ha we are applying resricions and defining ariables. BE CAREFUL WHEN USING A FORMULA! 33

34 Where do hese formulae come from? If acceleraion is consan, hen a = aerage acceleraion. a=(- 0 ) / ( 0) (a) () = (- 0 ) 1): = 0 + a If a = consan, hen elociy s ime graph is a sraigh line For a sraigh line graph: ae = (+ 0 )/2 Bu ae =(- 0 )/(-0) (- 0 )/(-0) = (+ 0 )/2 2): = 0 + (+ 0 )/2 Subsiue 1) ino 2) = a 2 /2 0 34

35 Eample: Les go back o our original eample of he car and assume ha he acceleraion is consan. We found ha a = 1.5 m/s 2 Les calculae he acceleraion. Recall 0 = 4 m/s, = 1 m/s, = 2s = 0 + a -1 m/s = -4 m/s + a(2 s) 3 m/s = a(2 s) a = 1.5 m/s 2 35

36 Freely Falling Objecs Near he earh s surface, he acceleraion due o graiy g is roughly consan: g = a Earh s surface = 9.81 m/s 2 oward he cener of he earh Free fall is he moion of an objec subjec only o he influence of graiy (no air resisance). An objec is in free fall as soon as i is released, wheher i is dropped from res, hrown downward, or hrown upward Quesion: Wha abou he mass of an objec? Answer: The acceleraion of graiy is he same for all objecs near he surface of he Earh, regardless of mass. 36

37 Graphical eample: A ball is hrown upward from he ground leel. = ball s heigh aboe he ground elociy is posiie when he ball is moing upward Why is acceleraion negaie? Is here eer deceleraion? 37

38 Preiously VECTORS Displacemen Aerage Velociy Insananeous elociy Aerage acceleraion Insananeous acceleraion SCALARS Disance Aerage speed Insananeous speed MOTION DIAGRAMS 38

39 eample 0 ime Draw s. and a s. 39

40 Consan acceleraion ( ) = + 0 a Velociy a = equals Velociy a =0 Plus he gain in elociy per second Muliplied by he ime span (eery second, he elociy increases Wih a m/s) 40

41 Consan acceleraion II ( 0 ( ) = + = + = + + a o o + 2 ) o o Sar posiion plus aerage speed muliplied by ime Subsiue = Subsiue ( ) = + 0 a 41

42 Free fall 5 kg 1 kg A B 100 m 1 2 ( ) = + + a o o ( ) = + 0 a 2 a is he he acceleraion fel due o graiaion (commonly called g=9.8 m/s 2 ) Why no mass dependence??? 42

43 (m/s) 2 =1,=2 (m/s) 2 =1,=2 0 0 T (s) 0 0 T (s) 1)Wha is he disance coered in 1 second? 2)Wha is he area indicaed by? Q a) b) c) The area under he - cure is equal o d) he displacemen of he objec! 43

44 Kinemaics in spors 44

45 100 m dash: wha is he bes sraegy? Afer long raining Ben Lewis can accelerae wih a=3.00 m/s 2 oer a disance of 20.0 m. Oer he remaining 80.0 m, he can mainain his op-speed. A) Afer how many seconds reaches Ben op-speed? B) Wha is his speed a ha ime? C) In how much ime does he cross he finish line? A) 20.0 m = 1/2*3.00* 2 =3.65 s B) (=3.65)=3.00*=10.95 m/s C) las 80 m: =80/10.95=7.30 s oal ime: =10.95 s 45

46 Afer a lo of raining... Ben manages o accelerae he firs 3.65 s wih a=4.00 m/s 2. Afer reaching his op-speed, he canno mainain i howeer, and slowly de-acceleraes (a=-0.4 m/s 2 ). Did his oal ime improe oer 100 m? A) Wha disance does Ben coer while acceleraing? B) Wha is his speed a ha ime? C) How long does is ake o coer o remaining disance and wha is his oal ime? A) =1/2*4.00* =26.6 m B) (=3.65 s)=4.00*3.65=14.6 m/s C) 100= *-1/2*0.4* *+372=0, so =5.51 or =67.5 oal ime: =

47 ???? Ben Lewis! 47

48 Problem Soling Sraegy Make a lis of gien quaniies Make a skech Draw coordinae aes idenify he posiie direcion Idenify wha is o be deermined Be consisen wih unis Check ha he answer seems reasonable Don panic. 48

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