CHAPTER THREE MOTION IN A STRAIGHT LINE

Transcription

1 PHYSICS Moion In ne Dimenion j k CHPTER THREE MTIN IN STRIGHT LINE. INTRDUCTIN Moion n objec i aid o be in moion, if i poiion chane wih repec o ime. Thi i relaed o he oberer. If i poiion i no chanin, he objec i aid o be a re. n objec in moion for one oberer may no be o for anoher oberer. The branch of phyic which deal wih he dy of moion of objec i called mechanic. I i diided ino hree: () Saic - Sdy of objec a re nder he acion of force in eqilibrim. () Kinemaic - Sdy of objec in moion wiho coniderin he cae of moion. () Dynamic - Sdy of he cae of moion. We are dealin wih kinemaic in hi chaper. We will inrodce a few phyical qaniie which are reqired o decribe he moion of an objec. Broadly peakin, here are hree ype of moion. () Tranlaory moion: Here paricle coniin he objec hae ame diplacemen. E: Moion of a car alon a road. () Roaory moion : Here all he paricle coniin he objec moe alon a circlar pah. E: Roaion of earh on i own axi. () cillaory moion: To and fro moion. E: moion of he pendlm of a clock. Poiion. ny objec i iaed a poin and hree oberer from hree differen place are lookin for ame objec, hen all hree oberer will hae differen oberaion abo he poiion of poin and no one will be wron. Becae hey are oberin he objec from heir differen poiion. berer ay : Poin i m away in we direcion. berer B ay : Poin i m away in oh direcion. berer C ay : Poin i 5 m away in ea direcion. Therefore poiion of any poin i compleely expreed by wo facor: I diance from he oberer and i direcion wih repec o oberer. Tha i why poiion i characeried by a ecor known a poiion ecor. Le poin P i in a xy plane and i coordinae are (x, y). Then poiion ecor (r ) of poin will be xˆi yˆj and if he poin P i in a pace and i coordinae are (x, y, z) hen poiion ecor can be expreed a r xi ˆ yj ˆ zkˆ.. Re and Moion. If a body doe no chane i poiion a ime pae wih repec o frame of reference, i i aid o be a re. Mohammed.jihar@en.ae W S C N E 5m m m B

2 Moion In ne Dimenion nd if a body chane i poiion a ime pae wih repec o frame of reference, i i aid o be in moion. Frame of Reference : I i a yem o which a e of coordinae are aached and wih reference o which oberer decribe any een. paener andin on plaform obere ha ree on a plaform i a re. B when he ame paener i pain away in a rain hroh aion, obere ha ree i in moion. In boh condiion oberer i rih. B oberaion are differen becae in fir iaion oberer and on a plaform, which i reference frame a re and in econd iaion oberer moin in rain, which i reference frame in moion. So re and moion are relaie erm. I depend pon he frame of reference. Tree i a re Tree i in moion Plaform (Frame of reference) Moin rain (Frame of reference). Type of Moion. ne dimenional Two dimenional Three dimenional Moion of a body in a raih line i called one dimenional moion. When only one coordinae of he poiion of a body chane wih ime hen i i aid o be moin one dimenionally. e... Moion of car on a raih road. Moion of freely fallin body.. Paricle or Poin Ma. Moion of body in a plane i called wo dimenional moion. When wo coordinae of he poiion of a body chane wih ime hen i i aid o be moin wo dimenionally. e.. Moion of car on a circlar rn. Moion of billiard ball. Moion of body in a pace i called hree dimenional moion. When all hree coordinae of he poiion of a body chane wih ime hen i i aid o be moin hree dimenionally. e... Moion of flyin kie. Moion of flyin inec. The malle par of maer wih zero dimenion which can be decribed by i ma and poiion i defined a a paricle. If he ize of a body i neliible in comparion o i rane of moion hen ha body i known a a paricle. body (Grop of paricle) o be known a a paricle depend pon ype of moion. For example in a planeary moion arond he n he differen plane can be premed o be he paricle. In aboe conideraion when we rea body a paricle, all par of he body ndero ame diplacemen and hae ame elociy and acceleraion..5 Diance and Diplacemen. () Diance : I i he acal pah lenh coered by a moin paricle in a ien ineral of ime. (i) If a paricle ar from and reach o C hroh poin B a hown in he fire.

3 PHYSICS Moion In ne Dimenion j k Then diance raelled by paricle B BC 7 m C (ii) Diance i a calar qaniy. (iii) Dimenion : [M 0 L T 0 ] (i) Uni : mere (S.I.) B () Diplacemen : Diplacemen i he chane in poiion ecor i.e., ecor joinin iniial o final poiion. (i) Diplacemen i a ecor qaniy (ii) Dimenion : [M 0 L T 0 ] (iii) Uni : mere (S.I.) (i) In he aboe fire he diplacemen of he paricle C B BC C ( B) ( BC) ( B)( BC)co 90 o = 5m () If S, S, S... S n are he diplacemen of a body hen he oal (ne) diplacemen i he ecor m of he indiidal. S S S S... S () Comparion beween diance and diplacemen : (i) The manide of diplacemen i eqal o minimm poible diance beween wo poiion. So diance Diplacemen. (ii) For a moin paricle diance can neer be neaie or zero while diplacemen can be. (zero diplacemen mean ha body afer moion ha came back o iniial poiion) i.e., Diance > 0 b Diplacemen > = or < 0 (iii) For moion beween wo poin diplacemen i inle aled while diance depend on acal pah and o can hae many ale. (i) For a moin paricle diance can neer decreae wih ime while diplacemen can. Decreae in diplacemen wih ime mean body i moin oward he iniial poiion. () In eneral manide of diplacemen i no eqal o diance. Howeer, i can be o if he moion i alon a raih line wiho chane in direcion. (i) If r and r B are he poiion ecor of paricle iniially and finally. Then diplacemen of he paricle r r r B B and i he diance raelled if he paricle ha one hroh he pah PB. n Y r B B r B r X.6 Speed and Velociy. () Speed : Rae of diance coered wih ime i called peed. Mohammed.jihar@en.ae

4 Moion In ne Dimenion (i) I i a calar qaniy hain ymbol. (ii) Dimenion : [M 0 L T ] (iii) Uni : mere/econd (S.I.), cm/econd (C.G.S.) (i) Type of peed : (a) Uniform peed : When a paricle coer eqal diance in eqal ineral of ime, (no maer how mall he ineral are) hen i i aid o be moin wih niform peed. In ien illraion moorcycli rael eqal diance (= 5m) in each econd. So we can ay ha paricle i moin wih niform peed of 5 m/. Diance Uniform Speed 5m 5m ec ec ec ec ec 5m/ 5m/ 5m 5m 5m 5m/ 5m/ 5m/ 5m m/ 5m/ (b) Non-niform (ariable) peed : In non-niform peed paricle coer neqal diance in eqal ineral of ime. In he ien illraion moorcycli rael 5m in econd, 8m in nd econd, 0m in rd econd, m in h econd ec. Therefore i peed i differen for eery ime ineral of one econd. Thi mean paricle i moin wih ariable peed. Diance Variable Speed 5m 8m ec ec ec ec ec 5m/ 8m/ 0m m 6m 0m/ m/ 6m/ 7m ec 7m/. VERGE VELCITY ND VERGE SPEED (c) erae peed : The aerae peed of a paricle for a ien Ineral of ime i defined a he raio of diance raelled o he ime aken. erae peed Diance raelled ; aken a aerae peed : When paricle moe wih differen niform peed ime ineral a,, Toal diance coered Toal ime elaped,... ec repeciely, i aerae peed oer he oal ime of jorney i ien a d d d =......,,... ec in differen Special cae : When paricle moe wih peed po half ime of i oal moion and in re ime i i moin wih peed hen a Diance aeraed peed : When a paricle decribe differen diance d, d, d,... wih differen ime ineral,,,... wih peed,,... repeciely hen he peed of paricle aeraed oer he oal diance can be ien a

5 PHYSICS Moion In ne Dimenion j k 5 a Toal diance coered Toal ime elaped d d d d d d d d... d... When paricle moe he fir half of a diance a a peed of and econd half of he diance a peed hen a Mohammed.jihar@en.ae When paricle coer one-hird diance a peed, nex one hird a peed and la one hird a peed, hen a. INSTNTNEUS VELCITY ND SPEED (d) Inananeo peed : I i he peed of a paricle a pariclar inan. When we ay peed, i ally mean inananeo peed. The inananeo peed i aerae peed for infinieimally mall ime ineral (i.e., Inananeo peed lim 0 d d 0 () Velociy : Rae of chane of poiion i.e. rae of diplacemen wih ime i called elociy. (i) I i a calar qaniy hain ymbol. (ii) Dimenion : [M 0 L T ] (iii) Uni : mere/econd (S.I.), cm/econd (C.G.S.) (i) Type ). Th (a) Uniform elociy : paricle i aid o hae niform elociy, if manide a well a direcion of i elociy remain ame and hi i poible only when he paricle moe in ame raih line wiho reerin i direcion. For a body moin wih niform elociy, he diplacemen i direcly proporional o he ime ineral. (b) Non-niform elociy : paricle i aid o hae non-niform elociy, if eiher of manide or direcion of elociy chane (or boh chane). If he direcion or manide or boh of he elociy of a body chane, hen he body i aid o be moin wih non-niform elociy (c) erae elociy : I i defined a he raio of diplacemen o ime aken by he body Diplaceme n r erae elociy ; a aken (d) Inananeo elociy : Inananeo elociy i defined a rae of chane of poiion ecor of paricle wih ime a a cerain inan of ime.

6 6 Moion In ne Dimenion Inananeo elociy lim 0 r dr d () Comparion beween inananeo peed and inananeo elociy (a) inananeo elociy i alway anenial o he pah followed by he paricle. When a one i hrown from poin hen a poin of projecion he inananeo elociy of one i elociy of one i repeciely., a poin he inananeo, imilarly a poin B and C are and Direcion of hee elociie can be fond o by drawin a anen on he rajecory a a ien poin. (b) paricle may hae conan inananeo peed b ariable inananeo elociy. Example : When a paricle i performin niform circlar moion hen for eery inan of i circlar moion i peed remain conan b elociy chane a eery inan. (c) The manide of inananeo elociy i eqal o he inananeo peed. (d) If a paricle i moin wih conan elociy hen i aerae elociy and inananeo elociy are alway eqal. (e) If diplacemen i ien a a fncion of ime, hen ime deriaie of diplacemen will ie elociy. Le diplacemen x 0 dx d Inananeo elociy ( 0 ) d d For he ien ale of, we can find o he inananeo elociy. Y B C X e.. for 0,Inananeo elociy and Inananeo peed (i) Comparion beween aerae peed and aerae elociy (a) erae peed i calar while aerae elociy i a ecor boh hain ame ni ( m/) and dimenion [ LT ]. (b) erae peed or elociy depend on ime ineral oer which i i defined. (c) For a ien ime ineral aerae elociy i inle aled while aerae peed can hae many ale dependin on pah followed. (d) If afer moion body come back o i iniial poiion hen 0 (a r 0 ) b 0 and finie a ( 0). a a

7 PHYSICS Moion In ne Dimenion j k 7 can be i.e. (e) For a moin body aerae peed can neer be neaie or zero (nle a 0 while a = or < 0. ) while aerae elociy.5 CCELERTIN The ime rae of chane of elociy of an objec i called acceleraion of he objec. () I i a ecor qaniy. I direcion i ame a ha of chane in elociy (No of he elociy) () There are hree poible way by which chane in elociy may occr When only direcion of elociy chane cceleraion perpendiclar o elociy When only manide of elociy chane cceleraion parallel or aniparallel o elociy When boh manide and direcion of elociy chane cceleraion ha wo componen one i perpendiclar o elociy and anoher parallel or ani-parallel o elociy e.. Uniform circlar moion e.. Moion nder raiy e.. Projecile moion () Dimenion : [M 0 L T ] () Uni : mere/econd (S.I.); cm/econd (C.G.S.) (5) Type of acceleraion : (i) Uniform acceleraion : body i aid o hae niform acceleraion if manide and direcion of he acceleraion remain conan drin paricle moion. Noe : If a paricle i moin wih niform acceleraion, hi doe no necearily imply ha paricle i moin in raih line. e.. Projecile moion. (ii) Non-niform acceleraion : body i aid o hae non-niform acceleraion, if manide or direcion or boh, chane drin moion. (iii) erae acceleraion : a a The direcion of aerae acceleraion ecor i he direcion of he chane in elociy ecor a a d (i) Inananeo acceleraion = a lim 0 d () For a moin body here i no relaion beween he direcion of inananeo elociy and direcion of acceleraion. Y Mohammed.jihar@en.ae a a a X

8 Poiio n 8 Moion In ne Dimenion e.. (a) In niform circlar moion = 90º alway (b) In a projecile moion i ariable for eery poin of rajecory. (i) If a force F d d x (ii) By definiion a d d ac on a paricle of ma m, by Newon nd law, acceleraion dx d F a m i.e., if x i ien a a fncion of ime, econd ime deriaie of diplacemen ie acceleraion (xi) cceleraion can be poiie, zero or neaie. Poiie acceleraion mean elociy increain wih ime, zero acceleraion mean elociy i niform conan while neaie acceleraion (reardaion) mean elociy i decreain wih ime. (xii) For moion of a body nder raiy, acceleraion will be eqal o, where i he acceleraion de o raiy. I normal ale i 9.8 m/ or 980 cm/ or fee/ a. If he elociy of a body chane eiher in manide or in direcion or boh, hen i i aid o hae acceleraion. b. For a freely fallin body, he elociy chane in manide and hence i ha acceleraion. c. For a body moin rond a circlar pah wih a niform peed, he elociy chane in direcion and hence i ha acceleraion. d. For a projecile, whoe rajecory i a parabola, he elociy chane in manide and in direcion, and hence i ha acceleraion. e. The acceleraion and elociy of a body need no be in he ame direcion. e : body hrown erically pward. f. If eqal chane of elociy ake place in eqal ineral of ime, howeer mall hee ineral may be, hen he body i aid o be in niform acceleraion.. Neaie acceleraion i called reardaion or deceleraion. h. body can hae zero elociy and non-zero acceleraion. E : for a paricle projeced erically p, elociy a he hihe poin i zero, b acceleraion i. k. If a body ha a niform peed, i may hae acceleraion. E : niform circlar moion l. If a body ha niform elociy, i ha no acceleraion..8 Poiion Graph. Drin moion of he paricle i parameer of kinemaical analyi (,, a, r) chane wih ime. Thi can be repreened on he raph. Poiion ime raph i ploed by akin ime alon x-axi and poiion of he paricle on y-axi. Le B i a poiion-ime raph for any moin paricle. y y D B y C x

9 PHYSICS Moion In ne Dimenion j k 9 Velociy = Chane in poiion aken y (i) y From rianle BC an BC C D y C.(ii) y P = 0 o o = 0 i.e., line parallel o ime axi repreen ha he paricle i a re. T P T = 90 o o = i.e., line perpendiclar o ime axi repreen ha paricle i chanin i poiion b ime doe no chane i mean he paricle poee infinie elociy. Pracically hi i no poible. P T = conan o = conan, a = 0 i.e., line wih conan lope repreen niform elociy of he paricle. P T i increain o i increain, a i poiie. i.e., line bendin oward poiion axi repreen increain elociy of paricle. I mean he paricle poee acceleraion. P i decreain o i decreain, a i neaie T i.e., line bendin oward ime axi repreen decreain elociy of he paricle. I mean he paricle poee reardaion. P conan b > 90 o o will be conan b neaie T i.e., line wih neaie lope repreen ha paricle rern oward he poin of reference. (neaie diplacemen). P B C Sraih line emen of differen lope repreen ha elociy of he body chane afer cerain ineral of ime. S T P T Thi raph how ha a one inan he paricle ha wo poiion. Which i no poible. P Mohammed.jihar@en.ae T The raph how ha paricle comin oward oriin iniially and afer ha i i moin away from oriin.

10 Velociy Diance 0 Moion In ne Dimenion By comparin (i) and (ii) Velociy = an = an I i clear ha lope of poiion-ime raph repreen he elociy of he paricle. Vario poiion ime raph and heir inerpreaion Noe : If he raph i ploed beween diance and ime hen i i alway an increain cre and i neer come back oward oriin becae diance neer decreae wih ime. Hence ch ype of diance ime raph i alid p o poin only, afer poin i i no alid a hown in he fire. For wo paricle hain diplacemen ime raph wih lope and poee elociie and repeciely hen.9 Velociy Graph. an an The raph i ploed by akin ime alon x-axi and elociy of he paricle on y-axi. Diance and diplacemen : The area coered beween he elociy ime raph and ime axi ie he diplacemen and diance raelled by he body for a ien ime ineral. Then Toal diance = ddiion of modl of differen area. i.e. d Toal diplacemen = ddiion of differen area coniderin heir in. i.e. r d here and are area of rianle and repeciely and i he area of rapezim. + cceleraion : Le B i a elociy-ime raph for any moin paricle cceleraion = From rianle BC, Chane in elociy aken an By comparin (i) and (ii) cceleraion (a) = an BC C (i) D C.(ii) I i clear ha lope of elociy-ime raph repreen he acceleraion of he paricle. y D B C x

11 Velociy Velociy Velociy Velociy Velociy Velociy Velociy Velociy PHYSICS Moion In ne Dimenion j k Vario elociy ime raph and heir inerpreaion = 0, a = 0, = conan i.e., line parallel o ime axi repreen ha he paricle i moin wih conan elociy. Velociy Velociy = 90 o, a =, = increain i.e., line perpendiclar o ime axi repreen ha he paricle i increain i elociy, b ime doe no chane. I mean he paricle poee infinie acceleraion. Pracically i i no poible. =conan, o a = conan and i increain niformly wih ime i.e., line wih conan lope repreen niform acceleraion of he paricle. increain o acceleraion increain i.e., line bendin oward elociy axi repreen he increain acceleraion in he body. decreain o acceleraion decreain i.e. line bendin oward ime axi repreen he decreain acceleraion in he body Poiie conan acceleraion becae i conan and < 90 o b iniial elociy of he paricle i neaie. Poiie conan acceleraion becae i conan and < 90 o b iniial elociy of paricle i poiie. Neaie conan acceleraion becae i conan and > 90 o b iniial elociy of he paricle i poiie. Neaie conan acceleraion becae i conan and > 90 o b iniial elociy of he paricle i zero. Neaie conan acceleraion becae i conan and > 90 o b iniial elociy of he paricle i neaie. Mohammed.jihar@en.ae

12 Velociy Velociy Velociy Velociy (m/ ) Speed Speed Speed Speed Moion In ne Dimenion Sample problem baed on elociy-ime raph Problem. ball i hrown erically pward. Which of he followin plo repreen he peed-ime raph of he ball drin i flih if he air reiance i no inored (a) (b) (c) (d) Solion : (c) Problem. In fir half of moion he acceleraion i niform & elociy radally decreae, o lope will be neaie b for nex half acceleraion i poiie. So lope will be poiie. Th raph 'C' i correc. No inorin air reiance mean pward moion will hae acceleraion (a + ) and he downward moion will hae ( a). The raph of diplacemen / ime i S I correpondin elociy-ime raph will be [DCE 00] (a) (b) (c) (d) V V V V Solion : (a) We know ha he elociy of body i ien by he lope of diplacemen ime raph. So i i clear ha iniially lope of he raph i poiie and afer ome ime i become zero (correpondin o he peak of he raph) and hen i will be neaie. Problem. In he followin raph, diance raelled by he body in mere i [EMCET 99] Y (a) 00 5 Solion : (a) (b) 50 (c) 00 (d) 00 Diance = The area nder raph () X S ( 0 0) 0 = 00 mere Problem. ball i hrown erically pward which of he followin raph repreen elociy ime raph of he ball drin i flih (air reiance i neleced) [CPMT 99; MU (En.) 000] (a) (b) (c) (d)

13 Velociy (m/ec) Veloci PHYSICS Moion In ne Dimenion j k Solion : (d) Problem 5. In he poiie reion he elociy decreae linearly (drin rie) and in neaie reion elociy increae linearly (drin fall) and he direcion i oppoie o each oher drin rie and fall, hence fall i hown in he neaie reion. c Which of he followin elociy ime raph i no poible (a) (b) (c) (d) Solion : (d) Paricle can no poe wo elociie a a inle inan o raph (d) i no poible. Problem 6. Velociy-ime raph of wo car which ar from re a he ame ime, are hown in he fire. Graph how, ha (a) Iniial elociy of i reaer han he iniial elociy of B (b) cceleraion in i increain a leer rae han in B (c) cceleraion in i reaer han in B (d) cceleraion in B i reaer han in Solion : (c) a cerain inan lope of i reaer han B ( B ), o acceleraion in i reaer han B B B Problem 7. Solion : (a) The adjoinin cre repreen he elociy-ime raph of a paricle, i acceleraion ale alon, B and BC in mere/ec are repeciely (a), 0, 0.5 (b), 0, 0.5 (c),, 0.5 (d), 0.5, 0 cceleraion alon m / 0 5 B (ec) 0 cceleraion alon B 0 0 cceleraion alon BC m/.0 Eqaion of Kinemaic. Thee are he ario relaion beween,, a, and for he moin paricle where he noaion are ed a : = Iniial elociy of he paricle a ime = 0 ec = Final elociy a ime ec a = cceleraion of he paricle Mohammed.jihar@en.ae

14 Moion In ne Dimenion = Diance raelled in ime ec n = Diance raelled by he body in n h ec () When paricle moe wih zero acceleraion (i) I i a nidirecional moion wih conan peed. (ii) Manide of diplacemen i alway eqal o he diance raelled. (iii) =, = [ a = 0] () When paricle moe wih conan acceleraion (i) cceleraion i aid o be conan when boh he manide and direcion of acceleraion remain conan. (ii) There will be one dimenional moion if iniial elociy and acceleraion are parallel or ani-parallel o each oher. (iii) Eqaion of moion in calar from Eqaion of moion in ecor from a a a a a.. a. ( ) a a n (n ) n (n ) () Imporan poin for niformly acceleraed moion (i) If a body ar from re and moe wih niform acceleraion hen diance coered by he body in ec i proporional o (i.e. ). So we can ay ha he raio of diance coered in ec, ec and ec i : : or : : 9. (ii) If a body ar from re and moe wih niform acceleraion hen diance coered by he body in nh ec i proporional o ( n ) (i.e. ( n ) n So we can ay ha he raio of diance coered in I ec, II ec and III ec i : : 5. (iii) body moin wih a elociy i opped by applicaion of brake afer coerin a diance. If he ame body moe wih elociy n and ame brakin force i applied on i hen i will come o re afer coerin a diance of n. a 0 a, a [ince a i conan] So we can ay ha if become n ime hen become n ime ha of preio ale. (i) paricle moin wih niform acceleraion from o B alon a raih line ha elociie a and B repeciely. If C i he mid-poin beween and B hen elociy of he paricle a C i eqal o and

15 PHYSICS Moion In ne Dimenion j k 5 Problem. Solion : (c) Problem. Solion : (d) Problem. Solion : (d) den i andin a a diance of 50mere from he b. oon a he b ar i moion wih an acceleraion of m, he den ar rnnin oward he b wih a niform elociy. min he moion o be alon a raih road, he minimm ale of, o ha he den i able o cach he b i (a) 5 m (b) 8 m (c) 0 m (d) m Le den will cach he b afer Similarly diance raelled by he b will be 50 a 50 To find he minimm ale of, hen = 0 m/. 50 d d ec. So i will coer diance. 0 ( a a m /, o we e = 0 ec for he ien condiion ) [KCET 00] car, moin wih a peed of 50 km/hr, can be opped by brake afer a lea 6m. If he ame car i moin a a peed of 00 km/hr, he minimm oppin diance i (a) 6m (b) m (c) 8m (d) m a 0 a ( a = conan) a m. The elociy of a blle i redced from 00m/ o 00m/ while raellin hroh a wooden block of hickne 0cm. The reardaion, amin i o be niform, will be [IIMS 00] (a) 0 0 m/ (b) 00 m /, 00 m /, 0. m 0 m/ (c).5 0 m/ (d) 5 0 m/ Problem. a (00 ) (00 ) m / body ar from re wih an acceleraion a. fer econd, anoher body B ar from re wih an acceleraion a. If hey rael eqal diance in he 5h econd, afer he ar of, hen he raio a : a i eqal o [IIMS 00] (a) 5 : 9 (b) 5 : 7 (c) 9 : 5 (d) 9 : 7 Solion : (a) By in n a S n, Diance raelled by body in 5 h econd = 0 a ( 5 ) Diance raelled by body B in rd econd i = 0 a ( ) ccordin o problem : 0 a ( 5 ) = 0 a a ( ) 9a 5a a Problem 5. The aerae elociy of a body moin wih niform acceleraion raellin a diance of.06 m i 0. m. If he chane in elociy of he body i 0.8m drin hi ime, i niform acceleraion i[emcet (Med.) 000] (a) 0.0 m (b) 0.0 m (c) 0.0 m (d) 0.0 m 5 9 Mohammed.jihar@en.ae

16 6 Moion In ne Dimenion Solion : (b) and Diance erae elociy cceleraion ec Chane in elociy m / Problem 6. paricle rael 0m in fir 5 ec and 0m in nex ec. min conan acceleraion wha i he diance raelled in nex ec (a) 8. m (b) 9. m (c) 0. m (d) None of aboe Solion : (a) Le iniial ( 0) elociy of paricle = for fir 5 ec of moion 0 5 a(5) for fir 8 ec of moion 0 8 a(8) 5 0 mere, o by in 5a 8 0 8a By olin (i) and (ii) mere m / a. (i)... (ii) m / Now diance raelled by paricle in oal 0 ec. 0 a by biin he ale of and a we will e a So he diance in la ec = m Problem 7. body rael for 5 ec arin from re wih conan acceleraion. If i rael diance S, S and in he fir fie econd, econd fie econd and nex fie econd repeciely he relaion beween Solion : (c) S S, S and S i m [MU (En.) 000] (a) S S S (b) 5S S S (c) S S S (d) S S S 5 5 Since he body ar from re. Therefore. S S S a(5) 5 a 0 00 (0 ) a 00 a S a S S = 75 a 5 (5 ) a 5 a S S a S S S = Th Clearly S S S 5 Problem 8. If a body hain iniial elociy zero i moin wih niform acceleraion 8 m / ec, he diance raelled by i in fifh econd will be (a) 6 mere (b) 0 mere (c) 00 mere (d) Zero Solion : (a) S n a(n ) 0 (8) [ 5 ] 6 mere Problem 9. Solion : (b) 5 a The enine of a car prodce acceleraion m/ec in he car, if hi car pll anoher car of ame ma, wha will be he acceleraion prodced [RPET 996] (a) 8 m/ (b) m/ (c) m/ (d) m / F = ma a if F = conan. Since he force i ame and he effecie ma of yem become doble m a m m, a m/ a m m a

17 PHYSICS Moion In ne Dimenion j k 7 Problem 0. body ar from re. Wha i he raio of he diance raelled by he body drin he h and rd econd. [CBSE PMT 99] (a) 7/5 (b) 5/7 (c) 7/ (d) /7 Solion : (a) S n ( n ),. Moion of Body Under Graiy (Free Fall). S S 7 5 The force of aracion of earh on bodie, i called force of raiy. cceleraion prodced in he body by he force of raiy, i called acceleraion de o raiy. I i repreened by he ymbol. In he abence of air reiance, i i fond ha all bodie (irrepecie of he ize, weih or compoiion) fall wih he ame acceleraion near he rface of he earh. Thi moion of a body fallin oward he earh from a mall alide (h << R) i called free fall. n ideal one-dimenional moion nder raiy in which air reiance and he mall chane in acceleraion wih heih are neleced. () If a body dropped from ome heih (iniial elociy zero) (i) Eqaion of moion : Takin iniial poiion a oriin and direcion of moion (i.e., downward direcion) a a poiie, here we hae = 0 a = + = [ body ar from re] [ acceleraion i in he direcion of moion] (i) h (ii) h (iii) h n (n )...(i) (ii) Graph of diance elociy and acceleraion wih repec o ime : h = 0 h h h a an = (iii) h = (/), i.e., h, diance coered in ime,,, ec., will be in he raio of : :, i.e., qare of ineer. (i) The diance coered in he nh ec, h n (n ) So diance coered in I, II, III ec, ec., will be in he raio of : : 5, i.e., odd ineer only. () If a body i projeced erically downward wih ome iniial elociy Mohammed.jihar@en.ae

18 8 Moion In ne Dimenion Eqaion of moion : h h h n (n ) () If a body i projeced erically pward (i) Eqaion of moion : Takin iniial poiion a oriin and direcion of moion (i.e., erically p) a poiie a = [ acceleraion i downward while moion pward] So, if he body i projeced wih elociy and afer ime i reache p o heih h hen (ii) For maximm heih = 0 So from aboe eqaion and =, h h ; h ; h ; h n (n ) h = 0 h h h (iii) Graph of diance, elociy and acceleraion wih repec o ime (for maximm heih) : ( /) a + (/ ) (/) + (/ ) a I i clear ha boh qaniie do no depend pon he ma of he body or we can ay ha in abence of air reiance, all bodie fall on he rface of he earh wih he ame rae. () In cae of moion nder raiy for a ien body, ma, acceleraion, and mechanical enery remain conan while peed, elociy, momenm, kineic enery and poenial enery chane. (5) The moion i independen of he ma of he body, a in any eqaion of moion, ma i no inoled. Tha i why a heay and lih body when releaed from he ame heih, reach he rond imlaneoly and wih ame elociy i.e., ( h / ) and h. (6) In cae of moion nder raiy ime aken o o p i eqal o he ime aken o fall down hroh he ame diance. of decen ( ) = ime of acen ( ) = /

19 PHYSICS Toal ime of flih T = + Moion In ne Dimenion j k 9 (7) In cae of moion nder raiy, he peed wih which a body i projeced p i eqal o he peed wih which i come back o he poin of projecion. well a he manide of elociy a any poin on he pah i ame wheher he body i moin in pward or downward direcion. (8) ball i dropped from a bildin of heih h and i reache afer econd on earh. From he ame bildin if wo ball are hrown (one pward and oher downward) wih he ame elociy and hey reach he earh rface afer and econd repeciely hen =0 (9) body i hrown erically pward. If air reiance i o be aken ino accon, hen he ime of acen i le han he ime of decen. > Le i he iniial elociy of body hen ime of acen and h a ( a) where i acceleraion de o raiy and a i reardaion by air reiance and for pward moion boh will work erically downward. For downward moion a and will work in oppoie direcion becae a alway work in direcion oppoie o moion and alway work erically downward. So h ( a) ( a) ( a) Comparin and we can ay ha > ince ( + a ) > ( a) ( a)( a) (0) paricle i dropped erically from re from a heih. The ime aken by i o fall hroh cceie diance of m each will hen be in he raio of he difference in he qare roo of he ineer i.e. m m m m = 0,( ), ( )...( ),.... Sample problem baed on moion nder raiy Problem. If a body i hrown p wih he elociy of 5 m/ hen maximm heih aained by he body i ( = 0 m/ ) [MP PMT 00] (a).5 m (b) 6. m (c).5 m (d) 7.6 m Mohammed.jihar@en.ae

20 0 Moion In ne Dimenion Solion : (a) Problem. (5 ) H max. 5 m 0 body fall from re in he raiaional field of he earh. The diance raelled in he fifh econd of i moion i ( 0 m / ) [MP PET 00] (a) 5m (b) 5m (c) 90m (d) 5m 0 hn n h5 h 5 If a ball i hrown erically pward wih peed, he diance coered drin he la econd of i acen i [CBSE 00] (a) (b) (c) ( ) (d) Solion : (b) Problem. Solion : (a) 5 m. If ball i hrown wih elociy, hen ime of flih elociy afer ec : So, diance in la '' ec : h. =. 0 ( ) ( ) h. Problem. man hrow ball wih he ame peed erically pward one afer he oher a an ineral of econd. Wha hold be he peed of he hrow o ha more han wo ball are in he ky a any Solion : (d) Problem 5. ime (Gien 9.8m / ) (a) lea 0.8 m/ (b) ny peed le han 9.6 m/ (c) nly wih peed 9.6 m/ (d) More han 9.6 m/ Ineral of ball hrow = ec. If we wan ha minimm hree (more han wo) ball remain in air hen ime of flih of fir ball m be U reaer han ec. i.e. T ec or ec 9.6 m / I i clear ha for 9. 6 Fir ball will j rike he rond (in ky), econd ball will be a hihe poin (in ky), and hird ball will be a poin of projecion or on rond (no in ky). man drop a ball downide from he roof of a ower of heih 00 meer. he ame ime anoher ball i hrown pide wih a elociy 50 meer/ec. from he rface of he ower, hen hey will mee a which heih from he rface of he ower [CPMT 00] (a) 00 meer (b) 0 meer (c) 80 meer (d) 0 meer Solion : (c) Le boh ball mee a poin P afer ime. Problem 6. The diance raelled by ball ( h )...(i) The diance raelled by ball B ( h)...(ii) By addin (i) and (ii) h h = 00 (Gien h h h 00. ) 00 / 50 8ec and h 0 m, h 80 m ery lare nmber of ball are hrown erically pward in qick cceion in ch a way ha he nex ball i hrown when he preio one i a he maximm heih. If he maximm heih i 5m, he nmber ec of ball hrown per mine i (ake 0m ) [KCET (Med.) 00] (a) 0 (b) 80 (c) 60 (d) 0 h ec P 00 m h h B Solion : (c) Maximm heih of ball = 5m, So elociy of projecion h ime ineral beween wo ball (ime of acen) = ec min m /

21 PHYSICS Moion In ne Dimenion j k Problem 7. Solion : (b) So no. of ball hrown per min = 60 paricle i hrown erically pward. If i elociy a half of he maximm heih i 0 m/, hen maximm heih aained by i i (Take m/ ) 0 (a) 8 m (b) 0 m (c) m (d) 6 m Le paricle hrown wih elociy and i maximm heih i H hen When paricle i a a heih From eqaion h H /, 0, hen i peed i H 0 m / H 00 Maximm heih H 00 0 = 0m Problem 8. Solion : (b) Problem 9. Solion : (b) one i ho raih pward wih a peed of 0 m/ec from a ower 00 m hih. The peed wih which i rike he rond i approximaely [MU (En.) 999] (a) 60 m/ec (b) 65 m/ec (c) 70 m/ec (d) 75 m/ec Speed of one in a erically pward direcion i 0 m/. So for erical downward moion we will conider 0 m / h ( 0) m / body freely fallin from he re ha a elociy afer i fall hroh a heih h. The diance i ha o fall down for i elociy o become doble, i [BHU 999] (a) h (b) h Le a poin iniial elociy of body i eqal o zero For pah B : 0 h (i) For pah C : () 0 x x (ii) Solin (i) and (ii) x h (c) 6 h (d) B C = 0 h x 8h Problem 0. body lidin on a mooh inclined plane reqire econd o reach he boom arin from re a he op. How mch ime doe i ake o coer one-forh diance arin from re a he op (a) (b) (c) (d) 6 Solion : (b) Problem. S a a conan / one dropped from a bildin of heih h and i reache afer econd on earh. From he ame bildin if wo one are hrown (one pward and oher downward) wih he ame elociy and hey reach he earh rface afer and econd repeciely, hen [CPMT 997; UPSET 00; KCET (En./Med.) 0 (a) (b) (c) (d) Solion : (c) For fir cae of droppin h. For econd cae of downward hrowin h Mohammed.jihar@en.ae

22 Moion In ne Dimenion ( )...(i) For hird cae of pward hrowin h ( )...(ii) on olin hee wo eqaion : Problem. By which elociy a ball be projeced erically downward o ha he diance coered by i in 5h econd. i wice he diance i coer in i 6h econd ( 0 m / ) (a) 58.8 m / (b) 9 m / (c) 65 m / 0 0 Solion : (c) By formla h n (n ) [ 5 ] { [ 6 ]} 5 ( 55) 65 m /. (d) 9.6 m / Problem. Waer drop fall a relar ineral from a ap which i 5 m aboe he rond. The hird drop i leain he ap a he inan he fir drop oche he rond. How far aboe he rond i he econd drop a ha inan [CBSE PMT 995] (a).50 m (b).75 m (c).00 m (d).5 m Solion : (b) Le he ineral be hen from qeion For fir drop () 5...(i) For econd drop 5 5 By olin (i) and (ii) x and hence reqired heih h 5.75 m. x Problem. balloon i a a heih of 8 m and i acendin pward wih a elociy of m /. body of k weih...(ii) i dropped from i. If 0 m /, he body will reach he rface of he earh in [MP PMT 99] (a).5 (b).05 (c) 5. (d) 6.75 Solion : (c) he balloon i oin p we will ake iniial elociy of fallin body m /, h 8m, By applyin h ; 8 (0 ) ec. 0 0 m / Problem 5. paricle i dropped nder raiy from re from a heih h( 9.8 m / ) and i rael a diance 9h/5 in he la econd, he heih h i [MNR 987] (a) 00 m (b).5 m (c) 5 m (d) 67.5 m Solion : (b) Diance raelled in n ec = Diance raelled in h n Solin (i) and (ii) we e. h. 5 m. n = h...(i) 9h ec (n )...(ii) 5 Problem 6. one hrown pward wih a peed from he op of he ower reache he rond wih a elociy. The heih of he ower i (a) / (b) / (c) 6 / (d) / Solion : (b) For erical downward moion we will conider iniial elociy =. 9 By applyin h, () ( ) h h.

23 PHYSICS Moion In ne Dimenion j k Problem 7. one dropped from he op of he ower oche he rond in ec. The heih of he ower i abo [MP PET 986; FMC 99; CPMT 997; BHU 998; DPMT 999; RPET 999] (a) 80 m (b) 0 m (c) 0 m (d) 60 m Solion : (a) h 0 80 m. Problem 8. body i releaed from a rea heih and fall freely oward he earh. noher body i releaed from he ame heih exacly one econd laer. The eparaion beween he wo bodie, wo econd afer he releae of he econd body i (a).9 m (b) 9.8 m (c) 9.6 m (d).5 m Solion : (d) The eparaion beween wo bodie, wo econd afer he releae of econd body i ien by : = ( ) = 9.8 ( ). 5 m. Mohammed.jihar@en.ae