() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration

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1 Secion 1.4 Moion in Spce: Velociy nd Acceleion We e going o dive lile deepe ino somehing we ve ledy inoduced, nmely () nd (). Discuss wih you neighbo he elionships beween posiion, velociy nd cceleion you emembe fom Clculus I nd II. In Spce ow we hve: () v() () posiion velociy cceleion veco veco veco () () () o v ( ) v( ) o () = v v () () ( ) = ( ) o ( ) () ( ) Speed of picle ime is he mgniude of he velociy veco: speed v( ) = Remembe v() = ()o ds d () = v () o dv d e of Δ of posiion wih espec o ime e of Δ of velociy wih espec o ime d = v + c v d = + c () () () () 1 HW #4: Find he velociy, cceleion, nd speed of picle wih he given posiion funcion. Skech he ph of he picle nd dw he velociy nd cceleion vecos fo he specified vlue of. = = ( ),4, 1

2 HW #16: Find he posiion veco of picle h hs he given cceleion nd he specified iniil velociy nd posiion. = i+ e j+ e k, v = k, = j+ k () ( ) ( ) Mos of his HW ssignmen is Modeling Pojeciles Wh s ewon s nd Lw? F = m (impessive) o shocus physics people!! Be ble o cee his. *A pojecile is fied wih n ngle of elevion α nd iniil velociy v. Assume i esisnce is negligible nd he only exenl foce is due o gviy. Repesen he posiion funcion () of he pojecile. = v cos α, v sinα g I will show you how we ge () ( ) ( ) x, hoizonl y, veicl If, by he wy, you elimine pmee you will see h y is qudic funcion of x. So he ph of he pojeciles is p of pbol. oice v is no veco bove bu i is below. y v α x d = hoizonl disnce of he pojecile

3 Foce due o gviy cs sigh downwd (negive) so i effecs he j no i. Acceleion due o he pull of gviy. = g j o, gviy o, g F = m F = m g j m g ( ) o, i,./lf. j Up/Down ms o f s Gviy is consn, consn cceleion, he book leves i g unil he vey end when you eplce i wih eihe he 9.8 o 3. o find he velociy veco, we mus inege he cceleion veco (cceleion due o gviy): v () = () = gj v( ) = g j + c1, c1 = v( ) = v iniil velociy v v = g() j + c = v ( ) So, v() = g j + v 1 his is he specific velociy veco once we found he consn c 1 nd subsiued v. o find he posiion veco, inege he velociy veco: () = v () = ( g j + v ) g () = j + v + c, c = ( ) =,, g () ( ) = j + v () + c =,, = + + c =,, ( ) A he oigin, so he iniil posiion is on he gound. g 1 () = j + v + c = g j + v +,, 1 () = g j + v Hee his is wien wih n iniil velociy veco, BU if we e given s n iniil speed wih cein ngle (which is wh we will un ino), hen

4 Given s n iniil speed, v (no longe veco bu scl), wih cein ngle, α. y v α x *If v = v d= hoizonl disnce of he pojecile (clled he nge of he pojecile) v(iniil speed of pojecile), hen he componens of veco = v cos α, v sinα 1443hoizonl veicl componen componen 1 vhen we cn wie () = g j + s 1 () = g j + vcos α,vsinα nd wiing i in he pefeed veco fom: 1 () =, g + vcos α, vsinα disibuing he : 1 () =, g + ( vcos α), ( vsinα) hen dding veco componens you ge: 1 () = + ( vcos α), g + ( vsinα) 1 = vcos α, v sinα g Finlly we hve * () ( ) ( ) x, hoizonl y, veicl he hoizonl disnce, d (he nge of he pojecile), is equl o he vlue of x when y = (oupu = ). So, genelly: 1 ( v sinα ) g = fco ou nd se boh fcos equl o 1 v sinα g = 1 = o v sinα g = v sinα = g his is he ime,, when he pojecile will ech d. When we plug his vlue of ino he hoizonl componen we will hve genel fomul o find d.

5 ( v ) vsinα vv sinαcosα sin α d = x-componen = ( vcosα) = ( vcosα) = = g g g So his is n esy wy o find his hoizonl disnce d, clled he nge of he pojecile. * ( ) v sin α d = his eches is mximum vlue when sin α = 1 o α = 45 o. I mkes sense, you ll ge g he mos disnce if you fie you pojecile off 45. Rnge of he pojecile. ge he mos disnce Given n iniil velociy veco of you pojecile, if you equie cein disnce, d, hee e wo ngles o ge o ge i hee. ennis nlogy lob o hi bck cou line o smshing, kille Seen Willims foehnd. HW #5: A gun hs muzzle speed of 15 m/s. Find wo ngles of elevion h cn be used o hi ge 8 m wy.

6 HW#6: Be his bsebll 3 f. bove he gound owds he cene field fence (which is 1 f. high nd 4 f. fom home ple). he bll leves he b wih speed of 115 f/s n 5 bove he hoizonl. Is i home un? 1 Memoize () ( v cos α), ( v sinα) g = posiion veco by sing wih cceleion nd ineging nd I would like you o be ble o come up wih his Memoize ( ) v sin α d = used when he pojecile is fied fom he oigin, so no shifed veiclly. g

7 ngenil nd oml Componens of Acceleion his is nice pplicion hey hew in hee becuse hey use he do poduc, coss poduc nd you need o find mgniudes. '( ) "( ) '( ) "( ) = = '() () In sudying he moion of picle, we ofen look he cceleion veco,, in wo componens, one in he diecion of he ngen, (ngenil Componen of Acceleion), nd he ohe in he diecion of he noml, (oml Componen of Acceleion). diecion of moion poins in he diecion he cuve is uning = c1 + c = + c1, c consns hown side o side while diving. oml componen of cceleion = k v Cuvue imes he sque of he velociy. hink of being in c. A shp cuve in he od mens lge k vlue so he componen of cceleion pependicul o he moion,, is lge nd he poo pssenge is hown gins he c doo. High speed ound he un hs he sme effec. ngenil componen of cceleion, = v'. Since gives he diecion of moion of he picle whip lsh in c, fowd o bckwd. = v' whip lsh e (which we e well we of). his would be he oice hee s no B inoml veco hee. o me how n objec moves hough spce, is cceleion lwys lies in he osculing plne (plnes of o ). If you nsfom nd using ( ), '( ), "( ), '() "() = '() '() "() = () Ou exbook goes hough he poofs if you e ineesed.

8 HW #34: Find he ngenil nd noml componens of he cceleion veco. () = 1 +, = ' " () () '() = ' " ( ) ( ) ()

Motion on a Curve and Curvature

Motion on a Curve and Curvature Moion on Cue nd Cuue his uni is bsed on Secions 9. & 9.3, Chpe 9. All ssigned edings nd execises e fom he exbook Objecies: Mke cein h you cn define, nd use in conex, he ems, conceps nd fomuls lised below: