PHYSICS 151 Notes for Online Lecture #4

Transcription

1 PHYSICS 5 Noe for Online Lecure #4 Acceleraion The ga pedal in a car i alo called an acceleraor becaue preing i allow you o change your elociy. Acceleraion i how fa he elociy change. So if you ar fro re and began acceleraing a 3 /, a he end of your elociy i 3 /, a he end of your elociy i 6 /, a he end of 3 your elociy i 9 /, ec. We can define an inananeou acceleraion a a = li and we can define an aerage acceleraion. a = = Exaple 5-: When you ener he expreway, you hae o change your elociy. Le aue ha he rap i 3 long. A he ar of he rap, you re going abou 35 ph which i 6 /. I ake you 5 econd o ge o he end of he rap, a which ie you re going 3 / (abou 65 ph). Wha i he aerage acceleraion? 3 6 a = = = = Le ake a look a he uni here = So he anwer i +.56 /. Ak yourelf wheher hi i a reaonable anwer. Acceleraion eaure how fa he elociy i changing. Here, he elociy i increaing, o i ake ene ha he acceleraion i poiie. Exaple 5-: You re approaching a chool zone and hae o low down fro 6 / o /. You didn noice he police car and hae o ep on he brake quickly, aking he change in.5 econd. Wha i he acceleraion? a = = 6 = 5. =.

2 Noe ha he negaie ign ean ha he elociy i decreaing. We oeie call a negaie acceleraion a deceleraion. The inananeou acceleraion i defined iilarly o he inananeou elociy li a = Uniforly Acceleraed Moion There are any cae of inere in which he acceleraion i a conan. In o cae, we know he poiion or elociy a one ie and wan o find eiher poiion or elociy a a econd ie. We re going o ubcrip all of our aring paraeer wih a zero. So a he ar of ie (which we ll ake o be zero), he objec i a poiion x and ha elociy. We wan o find paraeer a a general ie, Sar a ie = diplaceen a = x elociy a = A oe laer ie, diplaceen a = x elociy a = The aerage elociy a any ie will be (fro our definiion of aerage elociy) x = x x x = = x = x x Noe ha for conan acceleraion, he aerage acceleraion i equal o he inananeou acceleraion. Fro he definiion of acceleraion a = = = a = Noe ha hi equaion i only good for he cae when he acceleraion i conan in ie! We can rearrange hi equaion by uliplying boh ide by o ge and, rearranging, we find a = = + a (conan acceleraion only!)

3 To figure ou he poiion, hing are a lile rickier. = x x or x= x + Noe ha our fir equaion ha he inananeou elociy and no he aerage elociy. When he acceleraion i conan, he aerage elociy i he idpoin beween he iniial and he final elociie. + x = = (conan acceleraion only) Again, noe ha hi i rue only for conan acceleraion! x= x + ue he aboe eq. + x = x + now plug in our expreion for ( ) + + a x = x + A lile algebra x= x + + a conan acceleraion only! Your book derie a fourh equaion, which I no going o pend ie deriing, bu which i ery ueful. You ll noice ha all hree of he equaion we e deried conain ie. There are going o be oe cae in which ie i no known. = + a( x x ) conan acceleraion only! For copleene and x are lef in he aboe dicuion a ariable. Howeer, 99% of he ie boh of he are zero. A ypical kineaic proble will inole 4 of he fie ariable (x,,,a, & ). You will know 3 of hee and will be aked o ole for a fourh. The 5 h ariable i no inoled and idenifying hi ariable i your be clue o help you idenify he proper equaion. Equaion Variable no inoled = + a x x + a = = x = + a = + ax We can handle proble where x i no zero, by leing ubiuing x x for x in he aboe forula.

4 Exaple 5-3: A car oe fro re wih a conan acceleraion of 3 /. a) Wha i i elociy afer b) How far ha he car oed afer? Soluion: KNOWN: a = 3 / = = UNKNOWN Picure: = x = = +x a = 3 / = =? x =? = +a = + 3 ( ) = 3 / x= + a x= + 3 x = 3 3 x= = 5 ( )( ) ( )( ) ( )( ) Exaple 5-4: The police find kid ark of lengh 8 a an acciden cene. If he peron wa raeling 4 / (abou 9 ph), wha wa he acceleraion? +x = x = = 4 / a =? =? x = 8 = / KNOWN: UNKNOWN x=8 a = 4 / = / Noe ha we don know in hi proble, nor do we need o know i. There only one equaion ha doen ue :

5 = + a( x x ) = + ax = ax a = x ( 4 ) a = (8 ) 6 a = 6 a = ( ) Doe he inu ign ake ene? Ye! The car i lowing down, o he acceleraion hould be negaie You Try I! A rocke bla off and oe raigh upward fro he launch pad wih conan acceleraion. Afer 3. he rocke i a a heigh of 8.. (a) Wha are he agniude and direcion of he rocke acceleraion? (b) Wha i i peed a hi ie?

6 Graph and Acceleraion The inananeou acceleraion i defined iilarly o he inananeou elociy a = li Thi ean ha hey hae he ae graphical inerpreaion alo: he inananeou acceleraion i he lope of he elociy-ie graph. If an objec i oing wih conan elociy, he elociy-ie graph will look like hi: The acceleraion in hi cae i zero, becaue = o = eerywhere. So when here i conan elociy, here i zero acceleraion. o elociy (/) ie () In he cae of f conan nonzero acceleraion, he elociy. ie graph will look like he graph a righ. Call he final elociy f. Recall ha he area underneah he cure equal he diplaceen. Thi i ju a righ riangle, o he area i gien by ½ (bae) (heigh) or ie () A = ½ f Thi will coe in handy nex ie. elociy (/) You Try I! Deerine he acceleraion and diance raeled during each of he four egen below.