Average & instantaneous velocity and acceleration Motion with constant acceleration

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Howard Brown
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1 Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission dedline = Mondy, Feb. BEGINNING of clss -D Kinemics (reiew) Agend for Tody Aerge & insnneous elociy nd ccelerion Moion wih consn ccelerion Oeriew Kinemics: Descripion of Moion Posiion nd displcemen elociy» erge» insnneous ccelerion» erge» insnneous Pge

2 Posiion s Time Plos Gies locion ny ime. Displcemen is chnge in posiion. Slope gies elociy. (m) Posiion =3, (3) = 3 Displcemen beween =5 nd =. =. m -. m = -. m -. m Aerge elociy beween =5 nd =. = - m / s = -.5 m/s -.5 m/s -3 3 Velociy s Time Plos Gies elociy ny ime. Are gies displcemen Slope gies ccelerion. elociy =, () = 3 m/s Displcemen beween = nd =3: = = o =: ½ (3m/s) ( s) =.5 m = o =3: (3m/s) ( s) = 6 m Aerge elociy beween = nd =3? = Chnge in beween =5 nd =3. = 7.5 m 3 (m/s) Aerge ccelerion beween =5 nd =3: = -5 m/s / ( s) = -.5 m/s m / 3s =.5 m/s - m/s 3 m/s = -5 m/s Pge

3 Accelerion s Time Plos Gies ccelerion ny ime. Are gies chnge in elociy (m/s ) Accelerion =, () = - m/s Chnge in beween = nd =. = =-3: = (3m/s )(s) = 6 m/s =3-: = (-m/s )(s) = - m/s + m/s Consider his... Is i possible for n objec o he posiie elociy he sme ime s i hs negie ccelerion? - Yes - No Emple: A moing rin (forwrd) h is slowing down o sop. If he elociy of some objec is no zero, cn is ccelerion eer be zero? - Yes - No Yes, n objec cn be moing consn elociy. 6 Pge 3

4 Lecure ACT If he erge elociy of cr during rip long srigh rod is posiie, is i possible for he insnneous elociy some ime during he rip o be negie? A - Yes B - No Drie norh 5 miles, pu cr in reerse nd drie souh miles. Aerge elociy is posiie. 7 Conceps & Clculions, A skydier is flling srigh down, long he negie y direcion. During he iniil pr of he fll, her speed increses from 6 o 8 m/s in.5 s. Which of he following is correc? A) >, > B) >, < C) <, > D) <, < correc y During ler pr of he fll, fer he prchue hs opened, her speed decreses from 8 o 6 m/s in s. Which of he following is correc? A) >, > B) >, < C) <, > correc D) <, < If speed is incresing, nd re in sme direcion. If speed is decresing, nd re in opposie direcion. 8 Pge

5 Drw y s Dropped Bll A bll is dropped from heigh of wo meers boe he ground. y 3 A 3 B 3 C - 3 D - 3 E Dropped Bll A bll is dropped from heigh of wo meers boe he ground. Drw s Drw s Drw s Pge 5

6 Tossed Bll A bll is ossed from he ground up heigh of wo meers boe he ground. And flls bck down y Drw s 3 A 3 B 3 C - 3 D - 3 E Tossed Bll A bll is ossed from he ground up heigh of wo meers boe he ground. And flls bck down Drw s Drw s Drw s Pge 6

7 Lecure ACT A bll is hrown srigh up in he ir nd reurns o is iniil posiion. During he ime he bll is in he ir, which of he following semens is rue? A - Boh erge ccelerion nd erge elociy re zero. B - Aerge ccelerion is zero bu erge elociy is no zero. C - Aerge elociy is zero bu erge ccelerion is no zero. D - Neiher erge ccelerion nor erge elociy re zero. V e = Y/ = (Y f Y i ) / ( f i ) = e = V/ = (V f V i ) / ( f i ) No since V f nd V i re no he sme! 3 (meers) Emple - - posiion s. ime (m/s) (seconds) Where is elociy zero? Where is elociy posiie? Where is elociy negie? Where is speed lrges? Where is ccelerion zero? Where is ccelerion posiie? elociy s. ime (seconds) Pge 7

8 Summry of Conceps kinemics: A descripion of moion posiion: your coordines displcemen: = chnge of posiion elociy: re of chnge of posiion erge : / insnneous: slope of s. ccelerion: re of chnge of elociy erge: / insnneous: slope of s. 5 Moion in dimension In -D, we usully wrie posiion s ( ). Since i s in -D, ll we need o indice direcion is + or. Displcemen in ime = - is = ( ) - ( ) = - some pricle s rjecory in -D 6 Pge 8

9 -D kinemics Velociy is he re of chnge of posiion Aerge elociy in he ime = - is: ( ) ( ) = rjecory V = slope of line connecing nd. 7 -D kinemics... Consider limi Insnneous elociy is defined s: d( ) ( ) = d so ( ) = slope of line ngen o ph. 8 Pge 9

10 -D kinemics... Accelerion is he re of chnge of elociy Aerge ccelerion in he ime = - is: ( ) ( ) = And insnneous ccelerion is defined s: d( ) d ( ) ( ) = = d d using d( ) ( ) = d 9 Recp If he posiion is known s funcion of ime, hen we cn find boh elociy nd ccelerion s funcion of ime! = ( ) = d d = d d = d d Pge

11 More -D kinemics We sw h = d / d In clculus lnguge we would wrie d = d, which we cn inegre o obin: ( ) ( ) = ( ) d Grphiclly, his is dding up los of smll recngles: () = displcemen -D Moion wih consn ccelerion High-school clculus: Also recll h Since is consn, we cn inegre his using he boe rule o find: = d = d = + Similrly, since = = n d d d d d = d = ( + = + n + = + cons n + we cn inegre gin o ge: )d + Pge

12 Recp So for consn ccelerion we find: = + + = + = cons 3 Lecure, Ac 3 Moion in One Dimension When hrowing bll srigh up, which of he following is rue bou is elociy nd is ccelerion he highes poin in is ph? () Boh = nd =. (b),, bu =. y (c) =,, bu. Pge

13 Lecure, Ac 3 Soluion Going up he bll hs posiie elociy, while coming down i hs negie elociy. A he op he elociy is momenrily zero. Since he elociy is coninully chnging here mus be some ccelerion. In fc he ccelerion is cused by griy (g g = 9.8 m/s ). (more on griy in few lecures) The nswer is (c) =,, bu. 5 Useful Formul = + Soling for : = = + + Plugging in for : = + + = ( ) 6 Pge 3

14 Alerne (Clculus-bsed) Deriion d d d = = d d d (chin rule) d = d = d d d = = d ( - ) = ( ) d ( = consn) = ( ) 7 For consn ccelerion: Recp: = + + = + = cons From which we know: = ( = ( + ) ) 8 Pge

15 Emple A cr is reling wih n iniil elociy. A =, he drier pus on he brkes, which slows he cr re of b o b =, = 9 Emple... A cr is reling wih n iniil elociy. A =, he drier pus on he brkes, which slows he cr re of b. A wh ime f does he cr sop, nd how much frher f does i rel? b =, = = = f, = f 3 Pge 5

16 Emple... Aboe, we deried: = + Relize h = - b Also relizing h = = f : find = - b f or f = / b 3 Emple... To find sopping disnce we use: = ( ) In his cse = f =, = nd = f = ( b ) f f = b 3 Pge 6

17 Emple... So we found h =, = f f b b Suppose h o = 65 mi/hr = 9 m/s Suppose lso h b = g = 9.8 m/s Find h f = 3 s nd f = 3 m 33 Tips: Red! Before you sr work on problem, red he problem semen horoughly. Mke sure you undersnd wh informion is gien, wh is sked for, nd he mening of ll he erms used in sing he problem. Wch your unis! Alwys check he unis of your nswer, nd crry he unis long wih your numbers during he clculion. Undersnd he limis! Mny equions we use re specil cses of more generl lws. Undersnding how hey re deried will help you recognize heir limiions (for emple, consn ccelerion). 3 Pge 7

18 Recp of his week s lecures Mesuremen nd Unis (Chper ) Sysems of unis (Te: -) Dimensionl Anlysis (Te: -) Conering beween sysems of unis (Te: -5) -D Kinemics (Chper ) Aerge & insnneous elociy nd ccelerion (Te: -, -) Moion wih consn ccelerion (Te: -3, -5) Emple cr problem 35 Pge 8

Physics 100: Lecture 1

Physics 100: Lecture 1 Physics : Lecure Agen for Toy Aice Scope of his course Mesuremen n Unis Funmenl unis Sysems of unis Conering beween sysems of unis Dimensionl Anlysis -D Kinemics (reiew) Aerge & insnneous elociy n ccelerion