i. and displacement - usna 4...ch 4 motion in two ... 1. solution: ... tip of a blade, at a radius...
TRANSCRIPT
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CH 4
Motion in two and three Dimensions
I. PositionandDisplacement:
A. Position:
1. Thepositionofaparticlecanbedescribedbyapositionvector,withrespecttoareferenceorigin.
B. DisplacementVector: r
1. DisplacementVector:Thedisplacementofaparticleisthechangeofthepositionvectorduringacertaintime.
r
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C. Exampleof2Dmotion
1. Solution
t = 15 sec
SO:
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2. Graphs
II. AverageVelocityandInstantaneousVelocity
A. Ifaparticlemovesthroughadisplacementofrinttime,thentheaveragevelocityis:
B. Inthelimitthatthettimeshrinkstoasinglepointintime,theaveragevelocityisapproachesinstantaneousvelocity.Thisvelocityisthederivativeofdisplacementwithrespecttotime.
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C. Fortheprevioussampleproblem,let’sfindthevelocityoftherabbitat15sec:
1. Solution
t = 15 sec
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2. Graph
III. AverageandInstantaneousAccelerations
A. Followingthesamedefinitionasinaveragevelocity,
B. Ifweshrinkttozero,thentheaverageaccelerationvalueapproachestotheinstantaccelerationvalue,whichisthederivativeofvelocitywithrespecttotime:
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C. Fortheprevioussampleproblem,let’sfindtheaccelerationoftherabbitat15sec:
1. Solution
t = 15 sec
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2. Graph
IV. ExampleProblem
A. AParticlemovesintheXYplane.Itscoordinatesvarywithtimeas3
3( ) 1.00 ( ) 32.0 ( )
m mx t t t
s s and 2
2( ) 5.00 ( ) 12.0 ( )
m my t t t
s s .
Findtheposition,velocity,andaccelerationatt=3sec.
1. Solution
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V. Nextclass(Friday),PROJECTILEMOTION.
Read pages 70‐75
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VI. PROJECTILEMOTION.
A. Aprojectileis
1. Aparticlemovingintheverticalplane
2. Withsomeinitialvelocity
3. Whoseaccelerationisalwaysfree‐fallacceleration(g)
B. Themovementofaprojectileisprojectilemotion,withtheonlyaccelerationequaltothefreefallacceleration,g.
C. Inprojectilemotion,thehorizontalmotionandtheverticalmotionare__________________________________________;thatisneithermotionisaffectedbytheother.
D. Theinitialvelocityoftheprojectileis: 0v
1. Here, 0xv and 0 yv
Examples in sports:
Tennis
Baseball
Football
Lacrosse
Racquetball
Soccer………….
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E. Projectilemotionanalyzed,assumingnoexternalforcesotherthantheweight:
1. Followmyformatanditwillmaketheseproblemssimpler.
2. Startbyalwaysdrawingthefollowinggrid
Vfy = V0y + DV V0y + ayt
Vertical Horizontal
ay= ‐g =‐ 9.8 m/s2 ax=0
Dy (height) = Dx (range) = Voy = VoSinq= Vox = VoCosq= Vfy =
t = t =
3. Firstrule:TimeistheONLYInterloper!NothingelsecancrossbetweenVerticalandHorizontal
4. Thenlet’sfillitinwithourbig5equationsontheleftabovethewordvertical.
5. Afterwehavethebig5equationsontheleft,wewritethefirstequationontherightwithxsubscripts.Noticethatax=0sodrawalinethroughthe+½axt2becausethistermgoestozero.
6. Nowwillfillinourgivendatafromthewordproblem.
7. Thenwecirclewhichvariabletheywantustofind.
8. Thenwelookatourequationsandseewhichoneswewillusetonavigatethroughthisproblem.
We will do a sample problem shortly to make sure we understand this process. However, what if we wanted to plot the projectiles position or know its range. Let’s derive two equations which could help us determine these values more rapidly.
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F. Projectile’sPath
1. From
a) ThensolveEquation1fortime.
2. Andsince
3. PlugthatvalueoftbackintoEquationabove:
4. FinallywegettheEquationofPath.
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G. HorizontalRange,assumingnoexternalforces:
1. Thehorizontalrangeofaprojectileisthehorizontaldistancewhenitreturnstoitslaunchingheight
2. Thedistanceequationsinthex‐andy‐directionsrespectively:
3. Eliminatingt:
4. ThusR=
5. Caution:
6.
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H. Example,projectilemotion:
During the civil war, a cannon placed on top of a 40 m high hill is fired with a velocity of 200 m/s at an
angle of 30∞ from horizontal. How close did the cannon operators have to the enemy get in order to
ensure a hit?
1. Solution:
Vertical Horizontal
ay= ‐g =‐ 9.8 m/s2 ax=0
Dy (height) = Dx (range) = Voy = VoSinq= Vox = VoCosq= Vfy =
t = t =
Pleasant Hill – 40m
April 9, 1864
30 ±
200 m/s
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I. ExampleII,projectilemotion:
1. Uponspottinganinsectonatwigoverhangingwater,anarcherfishsquirtswaterdropsattheinsecttoknockitintothewater(Fig.below).Althoughthefishseestheinsectalongastraight‐linepathatangleanddistanced,adropmustbelaunchedatadifferentangle0ifitsparabolicpathistointersecttheinsect.If=36andd=0.900m,whatlaunchangle0isrequiredforthedroptobeatthetopoftheparabolicpathwhenitreachestheinsect?
a) Solution:∆∆
2
1/2
0
thus
and
2 thus
22
2
2 2
212
2 55.46°
Tuesday’s lesson (Monday’s a Holiday) will be Uniform Circular Motion read pages 76‐79
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VII. UniformCircularMotion
A. Aparticleisin“UniformCircularMotion”ifittravelsaroundacircleoracirculararcataconstant(uniform)speed.Note:althoughthespeedisnotchanging,theparticleisacceleratingbecausethevelocityischangingdirection.
B. Drawing
C. Centripetalacceleration(center‐seeking)
1. Proofisinthebookonpage77(proofdoesnotneedtobememorized).
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D. Period(orPeriodofrevolution)SymbolT.
1. Tisdefinedasthetimetomakeonerevolution.
2. T=
3. ac=
E. Side‐notes(willbeusefulwhenwegettochapter10,butonlyinterestingrightnow)
1. AngularVelocitynotmeterspersecond,butnumberofradianspersecond.Symbolw.w=2p/T
2. Velocity=w*r
3. ac=w2*r
4. Frequency=revolutionspersecond=1/T.UnitsHertzorrev/secorcyclespersec.
F. SampleProblem:UniformCircularMotion
1. Arotatingfancompletes1200revolutionseveryminute.Considerthetipofablade,ataradiusof0.15m.(a)Throughwhatdistancedoesthetipmoveinonerevolution?Whatare(b)thetip'sspeedand(c)themagnitudeofitsacceleration?(d)Whatistheperiodofthemotion?
a) Solution
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VIII. Relativemotioninone‐dimension
A. Thevelocityofaparticledependsonthereferenceframeofwhoeverisobservingthevelocity.
1. SupposeAlex(A)isattheoriginofframeA(asinFig.4‐18),watchingcarP(the“particle”)speedpast.
2. SupposeBarbara(B)isattheoriginofframeB,andisdrivingalongthehighwayatconstantspeed,alsowatchingcarP.Supposethattheybothmeasurethepositionofthecaratagivenmoment.Then:
wherexPAisthepositionofPasmeasuredbyA.
3. Graph
4. Consequently,
5. Also,
a) SincevBAisconstant,thelasttermiszeroandwehave
aPA = aPB.
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6. Example,relativemotion,1‐D:
a)
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Watch the following video about frames of reference it is very interesting and informative (it is a classic):
http://www.youtube.com/watch?v=Y75kEf8xLxI
IX. Relativemotionintwo‐dimensions
A. AandB,thetwoobservers,arewatchingP,themovingparticle,fromtheiroriginsofreference.BmovesataconstantvelocitywithrespecttoA,whilethecorrespondingaxesofthetwoframesremainparallel.rPAreferstothepositionofPasobservedbyA,andsoon.Fromthesituation,itisconcluded: