Kinematics in one and two dimensions (Cartesian coordinates only), projectiles; Relative velocity. to x f along x - axis at time t i

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1 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, PK K I N E M AT I C S Syllbs : CONCEPTS The prt of mechncs tht dels wth the descrpton of moton s clled knemtcs. There re two types of moton :. One dmensonl moton or Moton n strht lne : In ths moton the elocty ector nd ccelerton ector re lwys lon the sme lne.. Two dmensonl moton : In ths moton the elocty ector nd ccelerton ector wll be n the sme plne bt they re nclned t some nle, ths nle my chne drn the moton or my be constnt. For emple : crclr moton nd prbolc moton. C C Dsplcement nd Dstnce Dsplcement s defned s the chne n poston ector of the prtcle drn tme nterl wheres dstnce s defned s the lenth of ctl pth. Dsplcement s ector qntty wheres dstnce s sclr qntty. Note tht : ( dstnce dsplcement ( dstnce nd mntde of dsplcement re eql drn the tme nterl n whch the elocty of the prtcle shold not be zero t ny moment lon the strht lne moton. Velocty nd Speed Aere Velocty : The chne n poston ector.e. dsplcement dded by tme nterl drn whch ths chne occrs s known s ere elocty. For emple, prtcle chnes ts poston from to f lon - s t tme t nd t f respectely. Then ere elocty lon -s s en by : f. In enerl, for prtcle mon on cred pth : t t t f r rf t t f r t y z = î ĵ kˆ. Here r s the dsplcement drn the tme nterl t. t t t Instntneos Velocty : The elocty of the prtcle t prtclr pont or t prtclr nstnt of tme s clled the nstntneos elocty of the prtcle. It s en by r dr d dy dz lm = î ĵ kˆ = î y ĵ zkˆ. t 0 t dt dt dt dt For constnt elocty, dsplcement = (elocty (tme Aere Speed Knemtcs n one nd two dmensons (Crtesn coordntes only, projectles; Relte elocty. The ere speed of prtcle n tme nterl s defned s the dstnce trelled by the prtcle dded by the tme nterl.

2 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, Note tht : ( Aere Speed Aere Velocty ( ere speed & mntde of ere elocty re eql drn the tme nterl n whch the elocty of the prtcle shold not be zero t ny moment lon the strht lne moton. Instntnos Speed : The nstntneos speed eqls the mntde of the nstntneos elocty. The s ds nstntneos speed s en by lm t0 t dt speed s en by y z. where s s the dstnce trel drn tme t. Also the PK For constnt speed, dstnce = (speed (tme Clss Dscsson Problems :. A prtcle strts from one pont to nother pont lon the strht pth. It coers ths pth n n eql dstnce wth speed,... n. Fnd the ere speed for the complete jorney.. A prtcle strts from one pont to nother pont lon the strht pth. It coers ths pth n n eql tme nterl wth speed,... n. Fnd the ere speed for the complete jorney. 3. A prtcle s mon lon crclr pth of rds r. Fnd mntde of dsplcement nd dstnce for one qrter of crcle hlf crcle three qrter crcle complete one crcle (e.5 crcle. 4. A mn wlks on strht rod from hs home to mrket.5 km wy wth speed of 5 km h. Fnd the mrket closed, he nstntly trns nd wlks bck home wth speed of 7.5 km h. Wht s the mntde of ere elocty nd ere speed of the mn oer the nterl of tme ( 0 to 30 mn. ( 0 to 50 mn. ( 0 to 40 mn.? 5. The poston of prtcle mon on -s s en by = t 3 + 4t t + 4. Fnd the ere elocty of the prtcle between t = 0 to t = 4s. Home Work Problems : 6. A cr trels dstnce A to B t speed of 40 km/h nd retrns to A t speed of 30 km/h. C3 ( Wht s the ere speed for the whole jorney? ( Wht s the ere elocty? Answers : [( n n ( n n (3 R R, R, R 3 R R, 0, R (e R, 5R (4 ( 5km/h, 5km/h ( 0, 6 km/h ( 5/8 km/h, 45/8 km/h. (5 30 m/s (6 ( 34.3 km/h ( 0] Accelerton Aere Accelerton : Aere ccelerton s defned s the rto of chne n elocty to the tme tken. f where f nd re the elocty of the prtcle t t f (fnl tme nd t t t t (ntl tme respectely. f For strht lne moton (.e. lon -s f. t t t Instntneos Accelerton : Instntneos ccelerton s defned s f d Lm t0 t dt For strht lne moton (.e. lon -s d dt

3 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, Accelerton cn lso be epressed s d d d.. For nform elocty 0. d dt d PK 3 Unform ccelerton mens tht the ccelerton of the prtcle s constnt nd n ths cse < > =. If ccelerton s n sme drecton to the elocty then speed of the prtcle ncreses. If ccelerton s n opposte drecton to the leocty then speed decreses. Ths stton s clled retrdton. Note tht nete ccelerton does not men tht moton s retrdton. C4 Flow chrt to fnd dsplcement, elocty & ccelerton : Clss Dscsson Problems :. The poston of prtcle mon on -s s en by = t 3 + 4t t + 4. Fnd the elocty nd ccelerton of the prtcle t t = 4s.. A prtcle s mon lon the -s sch tht ts elocty, = where s constnt qntty. Proe tht the ccelerton of the prtcle s constnt. 3. A prtcle s mon lon the strht pth strtn wth elocty nd ccelerton t where s constnt nd t s the tme. Fnd the elocty t ny tme t nd dstnce trelled n tme t? 4. A prtcle s mon lon strht pth sch tht ccelerton =, where s the constnt nd s the nstnt elocty. If ntl elocty s then ( fnd elocty t ny tme t ( elocty fter coern the dstnce. Also fnd the mmm dstnce coered. Home Work Problems : 5. In the follown cses denotes the poston nd t denotes the tme. Fnd elocty nd ccelerton n ech cse nd lso drw the rph between elocty-tme nd ccelerton-tme : ( = t ( = t ( = t + bt ( = t + bt + ct 3 + d, where,b,c,d re poste constnts. 6. The elocty of prtcle mon on the -s s en by = + where n m/s nd s n m. Fnd ts ccelerton n m/s when pssn throh the pont = m. [Answers : ( 78 m/s, 3 m/s (3 = + t, s = t + 6 t 3 (4 ( = e t ( =, m. C5 dstnce = /(5 ( =, cc n = 0 ( = t, cc n = ( = + bt, cc n = b ( = + bt + 3ct, cc n = b + 6ct (6 30 m/s ] Grphcl Representton The dsplcement-tme, elocty-tme, ccelerton-tme nd other types of rphs re qte sefl n nlysn the moton of prtcle.

4 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com,. The ere elocty between two ponts A nd B s the slope of lne AB, wheres the nstntneos elocty of the prtcle t P s the slope of tnent drwn t ths pont s shown n fre. PK 4. Consder the elocty tme rph for prtcle mon lon the strht lne s shown n fre. Let the mntde of re of the trnle OAB s A nd BCD s A then Dstnce = A + A nd Mntde of dsplcement = A A 3. The ere ccelerton between two ponts A nd B s the slope of lne AB, wheres the nstntneos ccelerton of the prtcle t P s the slope of tnent drwn t ths pont. 4. On n ccelerton erss tme (t rph, the chne n elocty n elocty s the re bonded s shown n fre : Clss Dscsson Problems :. The elocty-tme rph for strht lne moton s shown n fre. Fnd totl dstnce totl dsplcement ere elocty between 5s to 40s totl ere speed (e totl ere elocty (f ere ccelerton between 5s to 5s ( ccelerton t t = 0 & 0s(h drw the ccelerton-tme rph, dstnce-tme rph nd dsplcement-tme rph.

5 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, Home work Problems :. Drw the poston-tme rph for nform elocty nform ccelerton nform retrdton frst nformly ccelerted nd then nformly deccelerted. 3. Drw the elocty-tme rph for nform elocty nform ccelerton nform retrdton frst nformly ccelerted nd then nformly deccelerted. 4. Drw the ccelerton-tme rph for nform elocty nform ccelerton nform retrdton frst nformly ccelerted nd then nformly deccelerted. [Answers : ( 90 m 0 m 85/35 m/sec 9/4 m/sec (e / m/sec (f 0.4 m/s ( /5 m/s, /5 m/s ] PK 5 C6 Strht Lne Moton wth Constnt Accelerton For nformly ccelerted moton lon strht lne (s -s the follown eqtons cn be sed. = ½ = + = 0 + (t t 0 = + ( 0 The symbols sed boe he follown menn; 0 Intl poston of the prtcle on -s t ntl tme t 0. Intl elocty of the prtcle lon -s. Velocty of the prtcle t ny poston nd ny tme t. Constnt ccelerton of the prtcle lon -s. NOTE : we mst decde t the bennn of problem where the orn of co-ordntes s nd whch drecton s poste. The choces of frme of reference re slly mtter of conenence. Clss Dscsson Problems :. A prtcle strts wth elocty lon strht lne pth wth constnt ccelerton. It ends ts jorney wth elocty. Fnd the elocty of the prtcle t the md pont of the jorney?. A body trels 00 cm n the frst two seconds nd 0 cm n the net for seconds. Fnd the elocty t the end of the seenth second from the strt? Home Work Problems : 3. A body, strtn from rest, moes n strht lne wth constnt ccelerton for tme nterl t drn whch t trels dstnce s. It contnes to moe wth the sme ccelerton for the net tme nterl t drn whch t trels dstnce s. Fnd the relton between s nd s? 4. A body mon n strht lne wth constnt ccelerton of 0 ms coers dstnce of 40 m n the 4th second. How mch dstnce wll t coer n the 6th second? C7 [Answers : ( Vertcl Moton Under Grty ( 0 cm/s (3 s = 3s (4 60 m] If body s mon ertclly downwrds or pwrds, t eperences downwrd ccelerton de to the rttonl force of the erth. Ths s clled ccelerton de to rty nd s denoted by the symbol. Strctly spekn s not constnt, bt res form plce to plce on the srfce of the erth nd lso wth heht. Howeer the rton of s so smll tht t cn be nelected nd cn be consdered constnt

6 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, nless ery lre hehts re noled. Therefore, we cn se the boe eqtons of moton for constnt ccelerton. For soln problems of ertcl moton nder rty, ether the pwrd or the downwrd drecton s tken s poste. If the pwrd drecton s tken s poste, then becomes nete nd ce-ers. The sns of other qnttes lke ntl elocty, ntl poston wll be decded ccordn to the frme of reference. Clss Dscsson Problems :. A prtcle s projected wth speed ertcl pwrd. Fnd tme to rech mmm heht tme to retrn ntl poston mmm heht reched?. A prtcle s dropped from rest from certn heht H boe the rond. Fnd tme to rech the rond the speed wth whch t wll rech the rond? 3. A prtcle s projected from mnr of heht 40m wth speed of 0 m/s ertcl pwrd. Fnd the tme of flht? 4. A stone s dropped from the top of 30 m hh clff. At the sme nstnt nother stone s projected ertclly pwrds from the rond wth speed of 30 m/s. The two stones wll cross ech other fter tme t nd the heht t whch they cross ech other s h then fnd t nd h. ( = 0 m/s 5. A prtcle, dropped from heht h, trels dstnce 9h/5 n the lst second. If = 9.8 m/s, then fnd the le of h? Home Work Problems : 6. A bll s thrown ertclly pwrds wth elocty of 0 m s from the top of mltstorey bldn. The heht of the pont from where the bll s thrown s 5.0 m from the rond. How hh wll the bll rse? How lon wll t trel before the bll hts the rond? Tke = 0 m s. 7. The dstnces trnsersed, drn eql nterls of tme, by body flln from rest, stnd to one nother n the sme rto s the odd nmbers bennn wth nty (nmely, : 3 : 5 : 7...]. Proe t. C8 [Answers : ( / / / ( (6 0 m 5 s] Moton n Plne or Two Dmensonl Moton H H (3 4s (4 s, 5m (5.5 m If prtcle s mon n plne, ts moton cn be splt nto two rectlner motons lon two perpendclr drectons. These two motons cn be treted ndependently of ech other nd then the reslts cn be combned ccordn to the rles of ector ddton & reqrement of the problem. Now, f the ccelerton s constnt, then the motons lon the two es re oerned by the follown two sets of eqtons : X-drecton Y-drecton = ½ y = y 0 + y + ½ y = + y = y + y = 0 + (t t 0 y y y = y 0 + (t t 0 = + ( 0 y = y + y (y y 0 Horzontl projecton Consder prtcle s projected horzontlly from certn heht h wth speed. Tkn ntl pont orn nd lon the drecton of projecton poste nd ertcl downwrds drecton poste, the moton wll be resoled ccordn to : PK 6

7 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, PK 7 X-drecton 0 = 0 y 0 = 0 = y = 0 = 0 Y-drecton y = Hence fter tme t : = t, = nd y = ½ t, y = t. Therefore, the mntde of the resltnt elocty fter tme t s t nd ts nclnton to the horzontl s en by t tn The horzontl dstnce trelled by the body n tme t s = t nd the ertcl dstnce s Combnn the boe two eqtons, we et y y t. Ths shows tht the trjectory of the prtcle s prbol. If T s the tme tken by the prtcle to rech the rond, then h T or T h The horzontl rne R s en by R T. Oblqe Projecton Consder prtcle s projected wth ntl elocty t n nle wth the horzontl s shown n fre. The boe moton wll be resoled ccordn to X-drecton Y-drecton 0 = 0 y 0 = 0 = cos y = sn = 0 y = h

8 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, Velocty fter Tme t : If nd y re the horzontl nd the ertcl components of elocty fter tme t, then = cos, y = sn t. The nle whch the elocty ector mkes wth the horzontl drecton t tme t s en by tn y sn t cos Poston fter Tme t : The horzontl nd ertcl dsplcement fter tme t re : PK 8 ( cost; y ( sn t Trjectory : Elmntn t from the boe two eqtons, we et the eqton of the trjectory of the projectle s y (tn cos t whch represents prbol. The follown mportnt reslts cn be esly obtned : Mmm Heht H sn Tme of Flht T Horzontl Rne R sn sn Two mportnt ponts to be noted concernn horzontl rne R : ( For en elocty of projecton, R s mmm when = ( For en elocty, there re two nles of projecton for whch the rne s the sme, f one of these nles s, the other s. Clss Dscsson Problems :. A prtcle strts from orn t t = 0 wth elocty 5.0îm / s nd moes n -y plne nder cton of force whch prodces constnt ccelerton of ( 3.0î.0ĵm / s. Wht s the y-coordnte of the prtcle t the nstnt when ts -coordnte s 84 m? Wht s the speed of the prtcle t ths tme?. A prtcle s projected wth speed t n nle of wth the horzontl. Another prtcle of dfferent mss s projected wth sme speed from the sme pont. Both the prtcles hs sme horzontl rne. Let the tme of flht nd mmm heht ttned by the frst prtcle nd second prtcle re t, h nd t, h respectely. Then fnd t /t nd h /h. 3. Let the mmm heht ttned by the projectle s n tmes the horzontl rne. Fnd the nle of projecton wth the horzontl? 4. A crcketer cn throw bll to mmm horzontl dstnce of 00 m. How mch hh boe the rond cn the crcketer throw the sme bll? 5. A fhter plne flyn horzontlly t n lttte of.5 km wth speed 70 km/h psses drectly oerhed n nt-rcrft n. At wht nle from the ertcl shold the n be fred for the shell wth mzzle speed 600 m s to ht the plne? At wht mnmm lttde shold the plot fly the plne to od ben ht? (Tke = 0 m s Home Work Problems : 6. A crcket bll s thrown t speed of 8 m s n drecton 30 0 boe the horzontl. Clclte the mmm heht. the tme tken by the bll to retrn to the sme leel, nd the dstnce from the thrower to the pont where the bll retrns to the sme leel.

9 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, 7. A hker stnds on the ede of clff 490 m boe the rond nd throws stone horzontlly wth n ntl speed of 5 m s. Nelectn r resstnce, fnd the tme tken by the stone to rech the rond, nd the speed wth whch t hts the rond. (Tke = 9.8 m s. [Answers : ( 36 m 6 m/s ( tn, tn (3 tn 4n (4 50 m (5 At n nle of sn (/3 = wth the ertcl; 6 km. (6 0 m.9 s 69 m (7 99 m/s] PK 9 C9 Relte Moton The poston, elocty nd ccelerton of prtcle re relte terms nd re defned wth respect to certn frme of reference. Ths frme of reference my be sttonry, mon wth constnt elocty or he some ccelerton. If AB s s poston of A wth respect to B then AB = A B where A nd B re the poston of A nd B wth respect to some common frme of reference. In the smlr wy for relte elocty AB = A B. In ector form r AB AB AB r A A A r B B B where r AB s the poston of A wth respect to B, r A nd r re the poston of A nd poston of B wth B respect to some common frme of reference, s the elocty of A wth respect to B, AB A nd B the elocty of A nd elocty of B wth respect to some common frme of reference, re s the ccelerton of A wth respect to B, nd re the ccelerton of A nd elocty of B wth respect to some common frme of reference. The eqtons of moton for constnt relte ccelerton re wrtten s : In enerl, A rel = ( 0 rel + rel + ½ rel rel = rel + rel rel rel = ( 0 rel + (t t0 rel = rel + rel ( rel ( 0 rel rel d dt Clss Dscsson Problems : rel, rel d dt B rel. On lon horzontlly mon belt, chld rns to nd from wth speed 9 km h (wth respect to the belt between hs fther nd mother locted 50 m prt on the mon belt. The belt moes wth speed of 4 km h. For n obserer on sttonry pltform otsde, wht s the speed of the chld rnnn n the drecton of moton of the belt? speed of the chld rnnn opposte to the drecton of moton of the belt?. Two prllel rl trcks rn north-soth. Trn A moes north wth speed of 54 km h, nd trn B moes soth wth speed of 90 km h. Wht s the elocty of B wth respect to A? elocty of rond wth respect to B?, nd elocty of monkey rnnn on the roof of the trn A nst ts moton (wth elocty of 8 km h wth respect to the trn A s obsered by mn stndn on the rond? 3. A bot cn trel t 0m/s relte to wter. It strts t one bnk of rer tht s 00 m wde nd flows wth elocty 7.5 m/s. If the bot ponts drectly cross, fnd ts elocty relte to the bnk. how fr downstrem t trels. AB

10 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, In whch drecton the bot mst pont so s to cross the rer n drecton perpendclr to the rer crrents. Also fnd the tme tken to cross the rer. 4. To mn wlkn t the rte of 3km/h the rn ppers to fll ertclly. When he ncreses hs speed to 6 km/h t ppers to meet hm t n nle of 45 0 wth ertcl from the front. Fnd the ctl speed of the rn? Home Work Problems : 5. A motorbot coers the dstnce between two sttons on the rer t = 8h nd t = h downstrem nd pstrem respectely. The tme tken by the bot to coer ths dstnce n stll wter s 6. A lft moes wth n ccelerton. A pssener n the lft drops book. The ccelerton of the book wth respect to the lft floor f the lft s on p nd f the lft s on down s respectely 7. A rlwy crre moes oer strht leel trck wth n ccelerton. A pssener n the crre drops stone. The ccelerton of the stone wth respect to the crre nd the Erth re respectely [( 3 km h ; 5 km h ; 0 s n ether drecton. ewed by ny one of the prents, the speed of the chld s 9 km h n ether drecton; nswer to s nltered ( 5 m s 5 m s 0 m s (3.5 m/s 75 m 6.6 m/s (4 3 km/h (5 9.6 h (6 +, (7, ] PK 0

11 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, PK E X E R C I S E TYPE (SINGLE CORRECT CHOICE TYPE. The nd y coordntes of prtcle t ny tme t re en by = 3t + 4t nd y = 4t where nd y re n m nd t n s. Then The ntl speed of the prtcle s 5 m/s. The ccelerton of the prtcle s constnt. The pth of the prtcle s prbolc. All re correct. Wter drops fll t relr nterls from roof. At n nstnt when drop s bot to lee the roof, the seprtons between sccesse drops below the roof re n the rto : : 3 : 4 : 4 : 9 : 6 : 3 : 5 : 7 : 5 : 3 : 3. A pont moes n -y plne ccordn to the lw = 4 sn 6t nd y = 4( cos 6t. The dstnce trersed by the prtcle n 4 seconds s ( nd y re n meters 96 m 48 m 4 m 08 m 4. A blloon strts rsn from the rond wth n ccelerton of.5 m/s. After 8 s, stone s relesed from the blloon. The stone wll coer dstnce of 40 m he dsplcement of 50 m rech the rond n 4 sec ben to moe down fter ben relesed 5. A body s n strht lne moton wth n ccelerton en by = 3 4. At t = 0 the elocty of the prtcle s 4 nt. The elocty when t = ln s 5/ 7/ 3/4 3/4 6. A stone s projected from the rond wth elocty of 50 m/s t n nle It crosses the wll fter 4s. The dstnce beyond the wll t whch the stone strkes the rond s 5 m 53 m 50 m 5/3 m 7. A stone s dropped from heht h, smltneosly, nother stone s thrown p from the rond whch reches heht 4h. The two stones cross ech other fter tme h h 8 8 h h 8. A cr, strtn from rest, s ccelerted t constnt rte ntl t ttns speed. It s then retrded t constnt rte ntl t comes to rest. The ere speed of the cr drn ts entre jorney s zero 9. The dstnce coered by body mon n strht lne n tme t s en by = t + t + 3. The ccelerton of the body wll ry s A projectle hs mmm rne of 500 m. If the projectle s now thrown p n nclned plne of 30 0 wth the sme elocty, the dstnce coered by t lon the nclned plne wll be bot 50 m 500 m 750 m 000 m. Three prtcles strts from the orn t the sme tme, one wth elocty lon the -s, the second lon the y-s wth elocty nd the thrd lon the = y lne. The elocty of the thrd so tht the three my lwys le on the sme lne s

12 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com,. The retest ccelerton or decelerton tht trn my he s. The mnmm tme n whch the trn cn et from one stton to the net t dstnce s s PK 6. A projectle s projected wth speed t n nle wth the horzontl. The tme fter whch the elocty ector of the prtcle become perpendclr to the ntl elocty of projecton s s s 3. A prtcle strts from the orn of coordntes t tme t = 0 nd moes n the y plne wth constnt ccelerton n the y-drecton. Its eqton of moton s y =. Its elocty component n the -drecton s rble s 4. Two prtcles A nd B re ntlly 40 m prt. A behnd B. Prtcle A strts mon wth nform elocty of 0 m/s towrds B. Prtcle B strtn from the rest hs n ccelerton of m/s n the drecton of elocty of A. The mnmm dstnce between the two s 0 m 5 m 5 m 30 m 5. A prtcle P s projected from the orn (0, 0 wth elocty of 0 ĵ t t = 0 nd nother prtcle Q s projected wth elocty from the orn t t = 5s. There s nform ccelerton on both prtcle lon the nete y-s. The prtcle P crosses the orn once n t t = 4s where s the prtcle Q crosses the pont ( 5, 0 t t = 9s. The elocty of the prtcle Q t t = 5s s 5 î 0ĵ 4 4 î 0 5 ĵ sn sn cos sn 7. A body s projected ertclly pwrds wth elocty. If t nd t be the tmes t whch t s t heht h boe the pont of projecton whle scendn nd descendn respectely, then t t, (t t h h tt, (t t h tt, (t t h tt, (t t 8. A prchtst drops freely from n eroplne for 0 s before the prchte opens ot. Then he descends wth net retrdton of.5 ms. If he bls ot of the plne t heht of 495 m nd = 0 ms, hs elocty on rechn the rond wll be.5 ms 7.5 ms 5 ms 0 ms 9. Two cyclsts moe towrds ech other. The frst cyclst, whose ntl elocty 5.4 km/h. descends the hll, thern speed wth n ccelerton of 0. m/s. The second cyclst whose ntl elocty 8 km/h clmbs the hll wth n ccelerton 0. m/s. The tme fter whch the cyclsts to meet f the dstnce seprtn them t the ntl moment of tme s 95 m s 30 s 35 s 5 s 40 s 0. A prtcle strts mon lon poste -s wth constnt ccelerton from orn. At tme t o fter the bennn of moton, the ccelerton reerses ts drecton remnn the sme n mntde. Tme t from the bennn of moton n whch the dsplcement of the prtcle becomes zero 4 î 0ĵ 5 none t 0 t 0 t 0 t 0 ( +

13 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, TYPE COMPREHENSION TYPE A prtcle moes n the -y plne wth constnt ccelerton drected lon the nete drecton of the y-s. The eqton of the trjectory of the prtcle s y = b where nd b re constnts.. The nle mde by the elocty ector wth the -s t the orn s tn tn b tn b b tn. The speed of the prtcle t the orn s b b b ( b 3. The poston lon the y-s t whch y s zero. TYPE 3 b b MATRIX-MATCH TYPE A projectle s projected wth certn speed t some nle wth the horzontl. Colmn-A 4b 8b Colmn-B (A The mmm rne for (P tn 4 s eql to (B The rne nd mmm (Q /4 heht re eql then = (C The nle between (R / elocty nd ccelerton t hhest pont (D The nle between (S /3 elocty nd ccelerton t the pont of projecton (T more thn / TYPE 4 MULTIPLE CORRECT CHOICE TYPE PK 3. A brd fles for 4 sec wth elocty of (t m/s n strht lne, where t = tme n seconds. Choose the correct sttements The ere speed s m/s The ere elocty s 0 The ere speed s ½ m/s The dstnce trelled drn s to 3s s m. For prtcles A, B, C nd D re thrown from the top of tower. A s thrown strht p wth speed, B s thrown strht down wth the sme speed, C s thrown horzontlly wth the sme speed nd D s relesed from rest. They ht the rond wth speed A, B, C nd D respectely nd tme of flht re t A, t B, t C nd t D respectely. Choose the correct sttements from the follown A = B = C D A B t D = t C D < A 3. Two prtcle re projected smltneosly n the sme ertcl plne from the sme pont, wth dfferent speeds nd, mkn nles nd respectely wth the horzontl. The pth followed by one, s seen by the other (s lon s both re n flht s ertcl strht lne f cos = cos strht lne f cos cos prbol hyperbol 4. A prtcle s thrown wth speed t n nle wth the horzontl. When the prtcle mkes n nle wth the horzontl, ts speed becomes. Then : the speed t the hhest pont s cos the speed t the hhest pont s cos = cos.sec = sec.cos 5. A rer s flown from west to est t speed of. A mn on the soth bnk of the rer, cpble of swmmn t wth respect to rer. The wdth of the rer s l. Choose the correct sttement. If the mn wnts to swm cross the rer n the shortest tme, he shold swm de north. If the mn wnts to swm cross the rer n the shortest dstnce, he shold swm de north. t t t

14 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, TYPE 5 If the mn wnts to swm cross the rer n the shortest dstnce, he shold swm sn north of west. ll re correct Asserton-Reson Type Ech qeston contns STATEMENT- (Asserton nd STATEMENT- (Reson. Ech qeston hs 4 choces (A, (B, (C nd (D ot of whch ONLY ONE s correct. (A (B (C (D Sttement- s Tre, Sttement- s Tre; Sttement- s correct eplnton for Sttement- Sttement- s Tre, Sttement- s Tre; Sttement- s NOT correct eplnton for Sttement- Sttement- s Tre, Sttement- s Flse Sttement- s Flse, Sttement- s Tre. STATEMENT- : Consder prtcle ntlly mon wth elocty of 5 m/s, strts decelertn t constnt rte of m/s lon -s. The dstnce trelled n the 3rd second s 0.5 m. STATEMENT- : The mntde of dsplcement s lwys eql to dstnce.. STATEMENT- : A prtcle P s projected wth elocty from the orn t n nle wth -s. Another prtcle s projected wth the sme elocty from the poston (l, 0 t n nle wth the -s. The locs of P s strht lne s seen from Q. STATEMENT- : Both prtcles he prbolc pth when they re n moton w.r.t. rond. 3. STATEMENT- : Two prtcles A nd B strt from rest nd moe for eql tme on strht lne. The prtcle A hs n ccelerton for the frst hlf of the totl tme nd for the second hlf. The prtcle B hs n ccelerton for the frst hlf nd for the second hlf. Prtcle A hs coered lrer dstnce. STATEMENT- : The elocty tme rph for the constnt ccelerton s strht lne wth non-zero slope. 4. STATEMENT- : d d dt dt STATEMENT- : For strht lne moton the ccelerton eqls to zero for constnt speed. PK 4 5. STATEMENT- : A prtcle projected wth speed t n nle of wth the horzontl n the ertcl plne. Another prtcle project wth the sme speed t n nle of wth the ertcl n the sme plne. They he the sme horzontl rne. STATEMENT- : They he the sme tme of flht. TYPE 6 (SUBJECTIVE TYPE PROBLEMS. A body trnsersed hlf the dstnce wth elocty 0. The remnn prt of the dstnce ws coered wth elocty for hlf of the tme, nd wth elocty for the other hlf of the tme. Fnd the men elocty of the body ered oer the whole tme of moton?. A projectle s projected wth elocty from the orn t n nle wth -s. Another prtcle B s projected wth the sme elocty from the poston (l, h, 0 t n nle wth the nete -s sch tht the two prtcles moe towrds ech other. Fnd the tme fter whch the seprton between the prtcles s mnmm. Also, fnd the mnmm dstnce between the prtcles. 3. A n shoots bllets tht lee the mzzle t 50 m/s. If the bllet s to ht tret 00 m wy t the leel of the mzzle, the n mst be med t pont boe the tret. How fr boe the tret s ths pont? 4. S prtcles stted t the corners of relr heon of sde l moe t constnt speed. Ech prtcle mntns drecton towrds the prtcle t the net corner. Fnd the tme the prtcle wll tke to meet ech other. 5. A cr strts mon rectlnerly, frst wth ccelerton = 5.0 m/s (the ntl elocty s eql to zero, then nformly, nd fnlly, decelertn t the sme rte, comes to stop. The totl tme of moton eqls t = 5 sec. The ere elocty drn tht tme s eql to = 7 km/hr. How lon does the cr moe nformly. 6. A bll s projected from orn wth n ntl elocty 0 = 700 cm/s n drecton 37 0 boe the horzontl s shown n fre. Another bll B 300 cm from orn on lne 37 0 boe the horzontl s relesed from rest t the nstnt A strts. ( = 9.8 m/s

15 Ensten Clsses, Unt No. 0, 03, Vrdhmn Rn Rod Plz, Vks Pr Etn., New Delh -8 Ph. : , 857, E-ml enstenclsses003@ml.com, How fr wll B he fllen when t s ht by A? In wht drecton s A mon when t hts B? 7. A ery brod eletor s on down ertclly wth constnt ccelerton ms. At the nstnt when the elocty of the lft s m/s, stone s projected from the floor of the lft wth speed of m/s relte to the floor t n eleton Fnd the tme tken by the stone to retrn to the floor. the rne of the stone oer the floor of the lft. [ = 0 ms ] 8. The crrent elocty of rer rows n proporton to the dstnce from ts bnk nd reches ts mmm le 0 n the mddle. Ner the bnk, the elocty s zero. A bot s so mon n the rer tht ts elocty relte to the wter s constnt nd perpendclr to the crrent. Fnd the dstnce throh whch the bot crossn the rer wll be crred wy by the crrent f the wdth of the rer s d. Also determne the trjectory of the bot? 9. A hnter s rdn n elephnt of heht 4m mon n strht lne wth nform speed of m/sec. He shts deer rnnn wth speed V n front t dstnce 45m mon perpendclr to the drecton of moton of the elephnt. If hnter cn throw hs sper wth speed of 0m/sec. relte to the elephnt, then t wht nle to t s drecton of moton mst he throw hs sper horzontlly for sccessfl ht. Fnd lso the speed V of the deer. 0. A prtcle s mon n plne wth elocty en by : 0î cost ĵ. If the prtcle s t the orn t t = 0. clclte the trjectory of the prtcle; fnd ts dstnce from the orn t tme (3/. d. c 3. TYPE (ANSWERS 4. c 5. d 6. b 7. b 8. d 9. c 0.. d. d 3. d 4. b 5. d d 8. c d TYPE.. d 3. b TYPE 3 [A-Q ; B-P ; C-R ; D-T]., b, d., b, c, d 3., b 4., b, c 5., c. C. B 3. D 4. D 5. C. 0 0 ( l., h cos m 4. TYPE 4 TYPE 5 TYPE 6 5. t t 4 / t 5 sec cm, horzontl sec, m d,y d d ;y 9. = 37 0, = 6m/s 0. y = sn (/ 0 0 [ (3 / 0 PK 5