Discussion Session 2 Constant Acceleration/Relative Motion Week 03

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1 PHYS 100 Dicuion Seion Conan Acceleraion/Relaive Moion Week 03 The Plan Today you will work wih your group explore he idea of reference frame (i.e. relaive moion) and moion wih conan acceleraion. You ll begin by viualizing ome iuaion uing diagram or graph o ge a feel for he concep. Then you ll olve hee problem uing he baic kinemaic idea you ve learned in PreLecure and praciced on your homework. Afer hi iniial pracice, your group will olve ome numerical and ome ymbolic problem o pracice repreening phyical relaionhip uing he conan acceleraion kinemaic idea. A he end, here a chance for your group o e your underanding by finding he miake() in an already worked oluion. 1

2 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 DQ1) Jim drive hi Viper 150 / norhward. Bill, who i 100 norh of Jim, drive hi Corvee ouhward a 100 /. When will hey mee? a) Talk wih your group o generae a kech of hi iuaion from he perpecive of omeone anding by he roadide (i.e. in he Earh reference frame ). Make ure you label all diplacemen and velociie. b) Now alk ogeher o generae a kech of hi iuaion from Jim perpecive (i.e. in Jim reference frame ). Again, make ure you label all diplacemen and velociie. To deermine he velociie in Jim frame, remember ha in Jim frame, Jim appear a re. To go from 150 / in he earh frame o re implie we ubrac Jim velociy from any velociy in he Earh frame vjim,jim = vjim,earh vjim,earh = = 0 v = v v = = 50 ( ) ( ) ( ) ( ) Bill,Jim Bill,Earh Jim,Earh c) In which frame doe he problem look impler? Dicu hi queion wih your group, wriing down ome of he facor ha go ino deciding. In Jim frame, only one objec i moving, while in he earh frame boh objec are moving. Jim frame make he moion appear impler. when your group ge here, call over he TA o check i ou before proceeding o he oluion* d) Working a a group, olve he problem in your choen imple frame of reference. How long doe i ake for he wo car o mee? In Jim frame, he Bill ravel 100 a 50 /, o hey hould mee in 100 = To ge hi yemaically from he kinemaic equaion, we would ue he known diplacemen, velociy, and acceleraion of Bill and olve for he ime: 1 x = x + v + a Bill 0,Bill 0,Bill Bill 1 ( 0 ) = ( 100 ) + ( 50 ) + ( 0 ) ( ) 100 = = =

3 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 If you were o olve hi problem in he Earh frame, you would need o wrie he poiion a a funcion of ime for each car, hen e heir poiion equal o find he ime ha hey mee: 1 1 x ( ) = x + v + a x ( ) = x + v + a Bill 0,Bill 0,Bill Bill Bill 1 ( 100 ) ( 100 ) ( 0 ) = + + ( 100 ) ( 100 ) x ( ) = Jim 0,Jim 0,Jim Jim Jim 1 ( 0 ) ( 150 ) ( 0 ) = + + ( 150 ) x ( ) = Seing he wo poiion equal allow u o find he meeing ime: x ( ) = x ( ) ( ) Jim Jim mee Bill mee v = d v Jim mee v + v = d Bill mee Bill mee d = = = = 0. 4 v + v mee Jim Bill 3

4 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 DQ) A he inan a raffic ligh urn green, a car ha ha been waiing a he inerecion ar moving a a conan acceleraion of 6 f/. A he ame inan a ruck i 10 fee behind he car, raveling wih a conan velociy of 0 f/. a) Working wih your group, plo he moion of he car and he ruck on he axe below. Aume x=0 i he iniial poiion of he car. Alo be ure o plo he moion over he full even econd. *once your group ha made he graph, call over he TA o check i ou * Thee plo came from uing he general diplacemen v. ime expreion for conan acceleraion kinemaic: Car Truck 1 1 xc ( ) = x + v + a x ( ) = x + v + a c0 c0 c 1 f ( 0) ( 0) ( 6 ) f ( 3 ) = + + = T T0 T0 T f 1 ( 10 m) ( 0 ) ( 0) f ( 10 m) ( 0 ) = + + = + The car graph hould herefore be a parabola ough he origin, while he ruck hould be a raigh line of lope 0 f/ going ough he poin (0 econd, -10 meer). You can hen plo he graph wih a few repreenaive poin 4

5 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 b) How many ime do he wo auomobile pa one anoher? A wha ime() doe hi occur? (Ue equaion o find he value hen check hem wih he graph) From he graph, i appear he wo auomobile pa each oher wice. We can ge hee ime from eing heir poiion equal and olving for he meeing ime: x ( ) = x ( ) c mee T mee f f ( 3 ) ( 10 ) ( 0 ) mee = f + f f ( 3 ) mee + ( 0 ) mee + ( 10 f) = 0 mee b b 4ac ± = a f f f ( 0 ) ( 0 ) 4( 3 )( 10 f) f ( 3 ) f f ( 0 ) ± ( ) 6 f mee ± = = = and 6. 1 Noe ha we had o ue he quadraic formula o olve he quadraic equaion reuling from eing he car and ruck poiion equal o one anoher. Thee wo ime are ju afer 0.5 econd and ju afer 6 econd. They agree nicely wih he inerecion in he plo above. c) Compare he velociie of he wo auomobile when hey pa. Check wih your group o ee if you all agree and if your anwer make ene in hi phyical iuaion. Phyically, we d expec he iniially faer-moving ruck o pa he car. Laer, he now faermoving car hould cach he ruck. We d herefore expec he ruck o have a faer peed a heir fir meeing, hen he car have a faer peed a heir econd meeing. The graph agree wih hi idea; he ruck lope (velociy) i eeper a heir fir meeing, bu he car lop i eeper a heir econd meeing. Numerically, he ruck velociy never change (alway 0 f/). The car velociy i given by ( 0) ( 6 f c c c ) v = v + a = +. The car peed a heir 0 meeing ime are herefore 3.8 f/ and 37 f/. A expeced, he car i going lower a heir fir meeing and faer a heir econd. 5

6 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 DQ3) A man i cruiing in hi por car a a peed of 10 fee/econd (abou 80 mph). He 480 fee away from a opligh when i urn red. Auming ha he maximum braking acceleraion ha hi car can apply afely i 0 fee/econd/econd, how much ime doe he have o noice he red ligh and ill op in ime? Talk wih your group o define ome ymbol ha repreen quaniie ha you are given or will need o olve hi problem and fill in he able below Symbol Meaning of Symbol Value (if known) vi vf amax xi xf xee max The iniial velociy of he car (when ligh urn red) The final velociy of he car (when i arrive a he ligh) The maximum value of he acceleraion of he car The iniial poiion of he car (when ligh urn red) The final poiion of he car (when i arrive a he ligh) The poiion of he car when driver noice he ligh The max ime beween ligh urn red and driver ee i and ill op a he ligh 10 f/ 0-0 f/ -480 f 0 Unknown. Noe ha we have choen a co-ordinae yem here which ha i origin a he opligh wih he iniial poiion negaive. Conequenly, he iniial velociy mu be poiive (o approach he ligh) and he acceleraion mu be negaive (in order o decreae he velociy o zero). In he pace below, wrie down he equaion uing hee ymbol ha will be needed o olve he problem Ue hi conan acceleraion equaion o deermine xee. Ue hi conan velociy equaion o deermine max. *once your group ha obained he equaion, call over he TA o check i ou * Solve hee equaion o find an expreion for max, he ime he ha o noice he red ligh and ill op in ime. 1 6

7 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 DQ4) Taiana ake 90 econd o walk eadily all he way up a alled ecalaor a a conan peed. Afer he ecalaor i fixed i acend a a conan peed, and he ride up he ame diance wihou walking in 60 econd. How much ime would i ake her o walk up he moving ecalaor if he walk a he ame rae up he moving ecalaor a he did when he ecalaor wa broken? Talk wih your group o define ome ymbol ha repreen quaniie ha you are given or will need o olve hi problem and fill in he able below Symbol Meaning of Symbol Value (if known) walk ride L vwalk vride vwalk+ride walk+ride The ime o walk up he alled ecalaor The ime o ride up moving ecalaor wihou walking The lengh of he ecalaor Taiana walking peed Taiana peed (wr ground) when riding wihou walking Taiana peed (wr ground) when riding wih walking The ime o walk up moving ecalaor Unknown. In he pace below, wrie down he equaion uing hee ymbol ha will be needed o olve he problem. We ar wih ee imple diance = rae ime ime equaion (conan velociy). We now add he equaion ha doe he work: he equaion of relaive moion when he i walking on he moving ecalaor: Taiana velociy wr ground = Taiana velociy wr elevaor + Taiana velociy wr ground *once your group ha obained he equaion, call over he TA o check i ou * We now have four equaion and five. Tha mean ha we canno olve for all of he. However, we have no more equaion o wrie! In hi cae, we do no need o know he value of all of he o deermine wha we wan: walk+ride. We will demonrae hi on he nex page. 7

8 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 Solve hee equaion o find an expreion for he ime i ake her o walk up he ecalaor when i i moving. We ar by ubiuing he equaion for each velociy ino he relaive moion equaion: Noe ha he L cancel here and we can olve for he ime direcly in erm of he wo known ime:

9 PHYS 100 Conan Acceleraion/Relaive Moion Week 03 Formula Shee Definiion Poiion Velociy Acceleraion x v = dx d a = = dv d d x d Conan Acceleraion v = v + a 0 x = x + v + a ( ) v = v + a x x 0 0 Relaive Moion va, B = va, E + ve, B v = v E, B B, E Conan and Converion m g = = 3 f 1 mile = Quadraic Formula If ± 4 ax + bx + c = 0 hen x = b b ac a 9

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