HORIZONTAL PROJECTILES
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- Clemence Carr
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1 HORIZONTAL PROJECTILES So wh i projecile? I i n objec h i lunched, hrown, or en ino he ir in uch w h i oe up/down nd/or forwrd he e ie. We will ue he -direcion when dicuing he up/down direcion, nd he -direcion when dicuing he horizonl, forwrd oeen. All projecile on he erh re ffeced b gri, which i cing downwrd in he direcion. We will lo ue in hi chper h here i no ir reince, which en h nohing low he projecile down in he forwrd direcion. So, wh doe he ph of projecile look like? Below re few eple, in order fro le difficul o nlze, o o difficul. The fir eple, which i ofen clled HORIZONTAL PROJECTILE, will be he fir eple h we ineige. In hi pe of projecile iuion, n objec i gien n iniil horizonl eloci. Thu, he eloci ecor i purel horizonl, nd lck -coponen. Thu, we r hee proble off b eing = 0. Ne, we nlze he oion eprel in he ericl nd horizonl direcion. To eplin wh, brief undernding of how hing fll igh be helpful. Look Figure #. Thi ie-lped picure how he filir free-flling objec, where i poiion fer eer econd of flling i hown. You cn ee h he objec i ccelering ince i relling longer dince in ech ucceie ie inerl. Ne, le look Figure #. Thi picure how n objec relling horizonll conn eloci. Since he dince relled during ech ucceie econd i he e, he ccelerion hould clerl be een being equl o zero k. conn peed. Figure #3 i cobinion of boh of hee oion. When projecile fll, i ccelere downwrd, nd hu i oion in he ericl direcion che h of Figure #. Howeer, ince nohing including ir low he objec down oe in he horizonl direcion, i oion in he horizonl direcion che h in Figure #. Thu, when ou pu boh oion ogeher, he projecile oion i prbolic in hpe. Figure #3 Figure # Figure #
2 Since projecile oion i clerl cobinion of ericl nd horizonl oion, we will nlze he oion eprel. The onl rible h he wo oion hre in coon i TIME, ince he oion in boh direcion hppen for he e oun of ie. When we nlze he oion in ech direcion, i i eie o keep he direcion full epred, nd hu we will ue chr hown below. Wh re here o n equion in he -direcion, nd o few in he -direcion? Thi i he reul of =0, which led o he eloci in he -direcion ing conn. Thu, here i onl one equion o ue in he -direcion, nd henceforh, we will cll hi equion SPLINTER. In he -direcion, on he oher hnd, we he he filir ccelerion of -9.8 /, nd hu we cn ue n of our 4 ninj urle equion. If ou forge heir ne, he re in order, LEONARDO, MICHAELANGELO, RAPHAEL, nd DONATELLO. Le look n eple of HORIZONTAL PROJECTILE, o ee how he work. Eple: A bll i rolled off fl roof h i 30 eer boe he ground. If he bll iniil peed i 5 /, find he ie needed o rike he ground well he dince h i lnd w fro he building ec ec ole for Horizonl Vericl conn f or Horizonl Vericl
3 We cn clerl ee h he ericl nd horizonl direcion were kep epred, like wo dodgebll e epred b he cener line. The onl rible h cn cro he line i ie, which would be he referee in our dodgebll nlog. Le r noher eple. Eple: A roo-beer ug lide off 4f ll ble. I lnd 9 fee w fro he ble. Wh w he peed of he ug when i lef he ble? Horizonl 9 f Vericl 4 f 3. ole for 0.498ec f 0 9 f /0.498ec 8.07 f / Cn we ALWAYS ue h = 0. NO!!!!!! Howeer, hi i lw rue of HORIZONTAL projecile, ince b definiion horizonl projecile he onl horizonl coponen o heir iniil eloci. Le r one l eple. Eple: A plne i fling wih peed of 00 / prllel o he ground. I fire bulle forwrd peed of 50 / relie o he plne. The bulle hi he ground 0 econd ler. Find he horizonl dince beween he oen h he bulle w fired nd he oen h i hi he ground. Alo, find he liude of he irplne. Horizonl 00 7,000 f 50 0 Vericl 9.8 ole for,960 herefore, 0 00 heliudei,960 Since he dince h we found in he preiou proble w dince relie o he ground, we needed o r he proble b finding he peed of he bulle relie o he ground
4 TRACKING PROJECTILES When we rck projecile we wn o know where he projecile i n oen in ie. We will lw ue h he projecile i he poin 0,0 when i r i fligh. To rck projecile, i ke ene o ue good book-keeping reg, or good reg o keep rck no pun inended of ll of our rible. I would highl recoend ble like he one hown in he eple proble below. We will keep rck of boh he projecile poiion, well i eloci cerin oen in ie. Since he eloci in he -direcion will be conn, bu he eloci in he -direcion will be increing negiel, ince i flling downwrd, he reuln eloci of he projecile will be geing ore lned ie progree. Thi cn be een in he picure below. The econd picure how he bll onl, while he fir picure reind u h he bll eloci i rell ecor RESULTANT of he horizonl nd ericl elociie ech oen in ie. When we li he direcion of eloci ecor, i i iporn o rei he epion o lw wrie he ngle [E S]. The projecile or no be oing e, bu i i o definiel no flling ouh. A reul, we will ue he er BELOW THE HORIZONTAL o gie direcion for crooked elociie. For he picure he righ, hi eloci would he direcion of o below he horizonl or BH. Since V = V = V, we lw know one ide of hi ringle righ fro he r. To find V, we ue n equion of oion in he - direcion. And o find he cul eloci of he objec, we dd he wo eloci coponen ecor, uing he Phgoren Theore o find he gniude nd inere ngen o ge he ngle. V V V
5 Eple: A bll roll off ble peed of 4 /. Trck i poiion,nd i eloci fer econd of flling. Horizonl 4 8 Vericl / V 9.6 / V = 0 / [78.46 o BH] Eple: A n on roller ke roll of fl cliff, which i high boe rier. Hi peed upon leing he cliff w 7 /. Trck hi poiion nd hi eloci ech of he fir 5 econd h he w flling. Tie = = ½ + = = = = 0 / 7 / 0 o BH 7 = = = = = = = = = = = = = = = Agin, noice h he ngle of he eloci ecor i geing lrger nd lrger ie progree, nd hu he eloci ecor i geing le horizonl nd ore ericl ie progree. Igine if he cliff in he l eple w infiniel ll. Eenull, he projecile will be flling lo enirel ericll downwrd, ince will be ooooooooo uch lrger hn.
6 In-Cl Eple: A bll i rolled off ble wih eloci of 5 /. If he ble i ll, find: i he ie he bll i in he ir. ii he dince w fro he ble h i lnd. A cr lide off he rod while king hrp urn. I lnd in rodide rine. If i lnd 0 fro he rod, nd i iniil eloci w 0 /, find he deph of he rine. 3 A hocke puck i lid cro 3 f ll ble. If he peron liding he puck wn o hi rge h i ling on he ground 0 f fro he ble, how f u he puck be liding when i lee he ble? 4 If n objec lide off 5 ll cliff wih peed of 5 /, find i ipc eloci. 5 An objec i lid off fl roofed, 50 ll building wih peed of 0 /. Trck i poiion : i = ec ii = ec iii = 3 ec 6 An objec i lid off 00 ll building whopping 00 /. Fir, find he ie needed o hi he ground. Then, rck i eloci : i ec ii ec iii 3 ec i 4 ec ipc Hoework Proble: 7. A CR ebll roll horizonll off of he edge of lunch ble. The ebll h n iniil eloci of / nd he ble i.5 eer boe he floor. How fr fro he be of he ble doe he ebll lnd horizonl rnge? 8. A 4 kg gllon of ilk i hrown horizonll / ou of 5 or-building window loced 3 eer fro he ground. Deerine he horizonl rnge of he ilk jug. 9. A brender lide ug long he op of br. I lide horizonll off of he -eer ll br urfce nd lnd eer fro he be. Deerine he originl eloci of he ug. 0. A bll roll wih peed of 0.6 /ec owrd he edge of ble h i 0.8 boe he floor. The bll roll off he ble. How long i i in he ir? How fr ou fro he ble doe he bll hi he floor?. Don Forge Throw Up : A bll i hrown righ up peed of 6 / fro heigh of 0.75 eer boe he ground. Find: he iu heigh reched boe he ground well he ie needed o hi he ground gin hink qudric!!!.. A lunch pil i ccidenll kicked off eel be on kcrper under conrucion. Suppoe he iniil horizonl peed i.0 /ec. Where i he pil poiion, nd wh i i peed nd direcion eloci 0.5 ec fer lunching? 3. Find he ipc eloci boh gniude nd direcion of he ebll in proble #7 boe. 4. Find he ipc eloci of he ug fro proble #9 boe. 5. A projecile i fired horizonll fro cnnon off er ll cliff. The projecile iniil eloci i 50 /. Trck: he poiion of he projecile =,, nd 3 econd. b he eloci of he projecile =,, nd 3 econd.
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