Kinemaics Review Ouline 1.1.0 Vecrs and Scalars 1.1 One Dimensinal Kinemaics Vecrs have magniude and direcin lacemen; velciy; accelerain sign indicaes direcin + is nrh; eas; up; he righ - is suh; wes; dwn; he lef Scalars have magniude nly disance; ; ime n signs 1.1.1A-B Disance and Displacemen Unis: meers; kilmeers; cenimeers; miles; fee; inches Displacemen is a sraigh-line pah frm he saring pin f a rip direcly an ending pin f ha rip. In 1D min direcin is saed (r shwn by sign) Disance is he al lengh f a pah raveled measured alng he acual pah, fllwing all f is urns and/r changes in direcin. There is n specific direcin (because i may change during rip!) Disance and lacemen can have he same magniude when min is in nly ne direcin. 1.1.1C-D Velciy and Speed sar end sar end disance = 3 cm lacemen = 3 cm eas r +3 cm end Unis: meers/secnd; miles per hur Velciy is defined as lacemen/ime. In 1D min direcin is saed (r shwn by sign) Speed is defined as al disance/al ime. May be calculaed fr any par f a rip Is fen calculaed as an average f w s There is n specific direcin (because i may change during rip!) Velciy and can have he same magniude when min is in nly ne direcin. sar ( = 0s) end ( = 1.5s) sar ( = 0s) end ( = 2.5s) = 2 cm/s sar velciy = 1.5 cm/s eas r +1.5 cm/s disance = 8 cm lacemen = 2 cm wes r -2 cm disance = 5 cm lacemen = 1 cm eas r +1 cm = 2 cm/s velciy = 0.4 cm/s eas r +0.4 cm end ( = 4s) sar ( = 0s) = 2 cm/s velciy = 0.5 cm/s wes r -0.5 cm/s
1.1.1E Accelerain Unis: meers/secnd 2 Accelerain is he rae a which /velciy changes. In 1D min direcin is saed (r shwn by sign) We fen refer magniude f accelerain his means ha we dn need knw he direcin, nly hw much accelerain here is. An bjec begins frm res and reaches a f 15 meers per secnd in 5.0 secnds. The average accelerain is 3 meers per secnd 2. 1.1.1F Kinemaics Equains Kinemaics equains are ls by which givens are used deermine an unknwn quaniy. Pssible givens: v avg ; v i ; v f ; a; d;. Wrie he equain, subsiue WITH UNITS, recrd answer WITH UNITS The phrase frm res means assume v i = 0 An bjec is drpped frm res near Earh s surface. Afer 2.5 secnds he bjec will be mving a 24.5 meers per secnd and will have fallen 30.7 meers. An bjec is hrwn upward (near Earh s surface) wih an iniial velciy f 12 meers per secnd. The bjec will reach is maximum heigh f 7.3 meers in 1.2 secnds. An bjec drpped frm a heigh f 2.0 meers n he Plane will reach he grund in 1.6 secnds while mving a a f 2.5 meers per secnd. This means ha he accelerain due graviy n Plane mus be 1.6 meers per secnd 2. An bjec mves wih an average f 10 meers per secnd. In 2.5 secnds, he bjec will have mved 25 meers. An bjec begins frm res and acceleraes a a rae f 2.0 meers per secnd 2 fr 6.0 secnds. The bjec will reach a final f 12 meers per secnd and will ravel 36 meers. 1.1.2 Freefall g = -9.81 m/s 2 near Earh s surface An bjec is nly cnsidered be in freefall AFTER i is released and under he influence f graviainal frce ONL. Apply any kinemaics equain wih he undersanding ha he accelerain is caused by graviy. Assume ha he siuain is near Earh s surface unless herwise saed. The phrase drpped means assume v i = 0 An bjec hrwn upward has + iniial velciy ; - accelerain ; and has a f zer a he p f is rip (maximum heigh). The upward and dwnward par f a rip are symmerical in erms f accelerain,, and ime.
1.1.3A-B One-Dimensinal Kinemaics Graphs Disance vs. Time NEVER decreases Read graph find curren al disance Subrac pins find disance raveled beween hem = slpe r Δd/Δ curve = changing (accelerain r decelerain) increasing slpe = increasing decreasing slpe = decreasing al disance raveled a = 3s is 4m disance raveled beween 0 and 8s is 7m disance raveled beween 6 and 9s is 2m Displacemen vs. Time Abve x-axis = he righ/eas/nrh/up frm rigin Belw x-axis = he lef/wes/suh/dwn frm rigin Read graph find curren psiin Difference beween pins = disance raveled Accumulae ge al disance = slpe r Δd/Δ psiive slpe = headed righ/eas/nrh/up negaive slpe = headed lef/wes/suh/dwn curve = changing (accelerain r decelerain) increasing slpe = increasing decreasing slpe = decreasing frm 2 4s is 0m/s frm 4 6s is 1m/s curren psiin a = 1s is +2 m curren psiin a = 4s = -2 m final psiin = 0 m dis. raveled beween 2 and 6s is 6 m dis. raveled beween 6 and 9s is 2 m al dis. raveled is 8 m dis dis dis dis avg. frm 3 5s is 0 m/s avg. frm 1 3s is 1.33 m/s avg. frm 2 6s is 2 m/s minless cns inc dec minless cns inc dec
Speed vs. Time NEVER ges belw zer Read graph find curren Average a w pins find v avg accelerain = slpe r Δv/Δ Area under graph = disance raveled a 3s is 2 m/s final is 0m/s is decreasing beween 7 and 8s avg. beween 0 and 4s is 1m/s avg. beween 7 and 8s is 2 m/s avg. accelerain beween 2 and 4s is 0 m/s 2 avg. accelerain beween 0 and 2s is 1 m/s 2 greaes accelerain beween 7 and 8s dis. raveled beween 0 and 4s is 6 m dis. raveled beween 5 and 7s is 8 m Velciy vs. Time Read graph find curren velciy Average velciy a w pins find v avg accelerain = slpe r Δv/Δ Areas under graph = disance/lacemen Areas abve x-axis are mvemen in + direcin Areas belw x-axis are mvemen in direcin ADD areas ge disance ADD + areas and SUBTRACT areas ge lacemen velciy a 1s is +2 m/s velciy a 4s is -2 m/s is increasing beween 0-1s and 3-4s is decreasing beween 2-3s and 7-8s avg. beween 0 and 1s is 0.5 m/s avg. beween 3 and 4s is 1 m/s avg. accelerain beween 1 and 2s is 0 m/s 2 avg. accelerain beween 0 and 1s is 2 m/s 2 dis. raveled beween 0 and 3s is 4 m dis. raveled beween 0 and 5s is 7 m. a 2s is +3m. a 8s is -4m minless cns inc dec minless cns inc. dec. cns inc. dec.
Accelerain vs. Time Read graph find curren accelerain 1.2.0 Tw Dimensinal Min - Cnsan Velciy in Bh Dimensins When bjecs mve in mre han ne dimensin a a ime (i.e. nrhwes), he min can be brken in w separae mins. Trignmery may be used break he prblem in w pars. Organize infrmain in / char. 1.2.0A-B Vecrs n acceleraing (culd be minless) cns inc. accel dec. accel 1.2 Tw Dimensinal Kinemaics Vecrs mus be added by arranging hem head--ail 3 cm 2.4 cm Use infrmain slve ne dimensin ime is he ONL variable ha can be used crss frm ne dimensin he her. Pyhagrean Therem can als be used find he resulan f w mins. A hckey puck mves nrh wih a f 6 meers per secnd. A he same ime, he puck drifs wes a a f 2 meers per secnd. In he amun f ime ha i akes he puck slide 24 meers nrh, i will als drif 8 meers wes. 2 m/s 6 m/s 24 m = 4 s d = 8 m = 4 s R 5.08 cm The nrhward cmpnen f a ba s velciy is 6 meers per secnd while he eas ward cmpnen f is velciy is 9 meers per secnd. The ba s resulan velciy is 10.8 meers per secnd. The ba will mve wih an angle ha is 33.7 nrh f eas. Vecrs can be added algebraically by finding and adding cmpnens A = 3 m a 20 B = 5 m a 160 A = 2.82 m B = -4.70 m R = -1.88 m A = 1.03 m B = 1.71 m R = 2.74 m R = 3.32 m a 124.5 R R R v res = 10.8 m/s 33.7 v eas = 9 m/s v nrh = 6 m/s
1.2.1A-F Grund Launched Prjeciles Prjeciles fllw parablic pahs. 1.2.1G-K Hriznally Launched Prjeciles Prjeciles fllw parablic pahs. v v max heigh v v d d d Break iniial velciy in cmpnens (if n given). d Organize verical and hriznal givens in a char. Make necessary assumpins a x = 0 (assume n air resisance) a y = -9.81 m/s 2 v ix = v i cs θ v iy = v i sin θ v y = 0 (a he maximum heigh) ime p = ½ al ime f fligh Fr greaes heigh and/r fligh ime a prjecile shuld be fired a 90. Fr greaes hriznal range a prjecile shuld be fired a 45. Prjeciles fired wih he same a cmplemenary angles will have he same hriznal range. A prjecile is fired wih an iniial velciy f 12 meers per secnd a an angle f 30. Breaking he prblem dwn we ge v i = 10.4 m/s v i = 6 m/s d =? v f = 0 (a p) d =? d =? Organize verical and hriznal givens in a char. Make necessary assumpins a x = 0 (assume n air resisance) a y = -9.81 m/s 2 v i = 0 v iy = iniial velciy ime is he same in bh dimensins Any w bjecs released frm he same heigh will hi he grund a he same ime regardless f saring hriznal velciy! A prjecile is fired hriznally wih an iniial velciy f 10 meers per secnd frm a heigh f 6 meers abve he grund. Breaking he prblem dwn we ge v i = 10 m/s v i = 0 m/s d =? d = 6 m = 1.1 s hriz. dis = 11 m p = 1.1 s = 1.21 s range = 12.7 m p = 0.61 s max h = 1.8 m
A prjecile is fired hriznally frm a heigh f 2 meers abve he grund and ravels 14 meers hriznally befre i lands. v i =? m/s v i = 0 m/s d = 14 m d = 2 m = 0.64 s v i = 21.9 m/s p = 0.64 s. A prjecile is fired hriznally wih a f 8 meers per secnd and ravels 16 meers hriznally befre i lands. v i = 8 m/s v i = 0 m/s d = 16 m d =? = 2 s p = 2 s d = 19.6 m