PHYS 154 Universi Phsics Laboraor Pre-Lab Spring 18 LAB 5 Projecile Moion CONTENT: 1. Inroducion. Projecile moion A. Seup B. Various characerisics 3. Pre-lab: A. Aciviies B. Preliminar info C. Quiz 1. Inroducion Afer inroducing one-dimensional moion he ne sep in our sud of moion is o allow bodies o move in a plane ha is perform wo-dimensional moions. In his case aking advanage from he proper of vecors is crucial inasmuch as kinemaic quaniies such as posiion veloci and acceleraion can be represened as arrows ling in he same veloci plane as depiced in he adjacen figure where I denoed r and respecivel v he posiion and he veloci of a car in he -plane of he road. Noe ha in a given coordinae ssem he vecor posiion is an arrow connecing he origin o he r locaion of he objec while he veloci is an arrow poining v alwas in he direcion of moion such ha i sas everwhere angen o he rajecor of he objec. In urn acceleraion a if an will have he direcion of whaever insananeous ne force acs on he objec changing is veloci depending on he paricular ineracions eperiences b he moving objec. Then since he vecor componen of all kinemaic quaniies will neal align along he respecive aes he sandard inerdependence beween posiion veloci and acceleraion can be disribued ino each componen: dr d d d v v v r v d v d d d d d (1) dv d dv dv a a a v v v v v v a d v a d d d d d. () The power of using vecor componens sas in ha he wo-dimensional problem involving vecors in general poining in differen direcions is reduced o wo one-dimensional problems involving componens which are scalars. Since vecors (magniude and direcion) are compleel characerized b heir componens b obaining he componens a an ime one obains he vecors a he respecive ime. In his lab uni we illusrae his vecor-based sraeg o model wo-dimensional moions using one eample: he rajecor of a projecile in is simples varian when i is overwhelmingl deermined b weigh.. Projecile moion Le us consider a projecile launched wih a cerain muzzle veloci (picall called iniial veloci). How can we undersand is ulerior rajecor in phsical erms? Noe ha if we neglec air resisance (a.k.a. drag) as soon as he objec leaves he launcher he onl pull eered on i is is own verical weigh. Hence is veloci will end o roae o poin downwards. Ye his canno happen insananeousl: he veloci will change direcion graduall while he projecile advances boh vericall and horizonall such ha he projecile will move along is characerisicall vauled 1
PHYS 154 Universi Phsics Laboraor Pre-Lab Spring 18 rajecor. Since he cause of his change of direcion is gravi i is given b graviaional acceleraion. Wih a modicum of curiosi ou ma summon wha we learned in class abou graphical vecor summaion o observe how graviaional acceleraion g is given b he change of veloci v v 1 beween an wo poins on he rajecor divided b he raveling ime Δ as shown in he adjacen diagram. You can pick up an wo veloci vecors because he average acceleraion in an ime inerval will be equal o he (approimael) consan graviaional acceleraion. To se off he analsis of projecile moion one needs o reformulae he quesion ha canalized ou discussions abou kinemaics: Knowing he iniial posiion and veloci of he projecile how can one predic is posiion and veloci a a laer ime? In he pas we answered his quesion for some one-dimensional moions b developing kinemaic equaions. Recall for insance he equaions for uniforml acceleraed moion which we applied o he verical free fall. In urn o answer he quesion in wo-dimensional siuaions one needs o emplo vecor properies more mehodicall. Thus neglecing he drag he equaions of moion can be immediael obained from he observaion ha he ne acceleraion is he graviaional such ha he ne acceleraion is a = (a a ) = ( g). Then he moion of a projecile launched a ime = from posiion r = ( ) wih iniial veloci v = (v v ) can be characerized b appling relaionships (1) and (): A. Seup v v v v a d v a d v v g (3) r v d v d v v g 1. (4) Hence o analze a free-fall kind of projecile moion one can follow a simple sep-b-sep sraeg: 1. Embed he wo-dimensional rajecor in a coordinae ssem for insance including horizonal and verical aes wih he projecile iniiall a coordinaes and.. Spli he iniial veloci v of he projecile ino - and -componens v and v which can be calculaed in erms of iniial speed v and iniial launching angle θ : v v cos (5) v v sin (6) 3. Noe ha if onl weigh acs on he projecile he acceleraion is vericall downward so he -componen of he veloci sas consan while he -componen is acceleraed uniforml. So we can calculae he componens of he posiion and veloci a an laer ime using he sandard equaions wih zero and hen consan acceleraion g: Along : v (7) v v (8) Along : 1 (9) v g v v g (1) 4. Once he componens a he laer ime are found one can calculae he wo-dimensional veloci and posiion using he formulas relaing he magniude and direcion of a vecor o is componens. For insance for he veloci a ime : Magniude: v v v (11) 1 Direcion: an v v (1) v 1 Δv v v 1 g = Δv Δ = v v 1 Δ
PHYS 154 Universi Phsics Laboraor Pre-Lab Spring 18 Eample: A projecile is launched wih iniial speed v = 98 m s a an angle θ = 3 above he horizonal as shown on he figure. Neglecing air resisance wha are he coordinaes and he veloci of he projecile 8seconds laer? (m) 15 1 ma 5 (m) θ 1 3 4 5 6 7 8 R 9 iniial = = laer ime The componens of he iniial veloci are: v 98 m/s cos3 85 m/s v 98 m/s sin 3 49 m/s Consequenl using equaions (3)-(6) he coordinaes and veloci are given b v 85 m/s 8. s 68 m 3. Pre-lab A. Aciviies v g 1 78.4 m 85 m/s v v v v v 85 m/s 9 m/s 87 m/s v 1 9 m/s v g 49 m/s 9.8 m/s 8. s 9 m/s an 19 85 m/s 1. Read carefull he inroducor maerial provided above. Make sure ha ou undersand: The principles ha shape up he rajecor of a projecile. The logic of using he equaions along horizonal and verical aes o predic he coordinaes and veloci of he projecile a an ime afer i is launched.. Answer he quesions on he quiz a he end of his documen. Noe ha he are based on observing a simulaion of projecile moion accessible via he course homepage. The quesions are also available on he Blackboard sie associaed wih he PHYS 154 lecure. B. Preliminar informaion The subjec of LAB 5 is Projecile Moion. Is main goal is o develop and help ou pracice our analical skills in suding he wo-dimensional moion of a projecile. You are designing he specificaions for a machine which mus hrow an objec from an elevaion (he lab able) across an area which ou do no have access o o a safe area below (on he floor). To illusrae he principles of our machine ou will make predicions abou he rajecor of he projecile using he kinemaic model developed in class (also discussed above) and hen emplo a ballisic pendulum as a projecile launcher o demonsrae he viabili of he model b esing he predicions Technical Commens: You will ackle he eperimen in wo seps: r 1. PART 1: Familiarize ourselves wih he launcher and measure he muzzle veloci of projeciles.. PART : Use he iniial posiion and veloci of he projecile o hi a arge placed a a locaion prediced b he heoreical model. 3
PHYS 154 Universi Phsics Laboraor Pre-Lab Spring 18 In none of he wo pars ou will measure he ime ha he projecile spends in he air. So he heoreical sraeg will be o combine equaions (3) and (5) above. Tha is subsiue he ime from equaion (5) o (3) hence obain relaionships beween verical and horizonal displacemens. In PART 1 he projecile will be fired horizonall from a desk on he floor (θ = ). You will measure he horizonal and verical displacemens and use hem o esimae he iniial veloci of he projecile. The necessar calculaion was performed in class (Problem 3). Recall ha we obained he ime from Eq. (5) for he verical displacemen Δ b seing v v = and hen subsiuing i in Eq. (3) for he horizonal displacemen Δ. PART 1 Review ha calculaion because ou will have o reproduce i on our lab repor. In PART he projecile will be fired wih a non-zero iniial angle from a desk on he floor (θ ). The iniial veloci of he projecile will be assumed known (from PART 1) as well as he verical displacemen (measured). In urn ou will have o predic and es he horizonal displacemen Δ. However o find he ime from Eq. (5) is a ad more challenging because now v = v sin θ so he ime is a soluion of he quadraic equaion: Δ (Eq.5) v θ Δ (Eq.3) PART 1 vsin g (13) measure PART 1 se Δ (Eq.5) This equaion has wo soluions bu one of hem is negaive so ou will need onl he posiive one which makes phsical sense: Δ (Eq.3) fligh v sin v sin g (14) g Noe ha when he projecile is fired a an iniial aliude h he verical displacemen is Δ = = h = h. Then our heoreical horizonal displacemen will be calculaed from v cos. (15) fligh Undersand and pracice hese calculaions because ou will have o reproduce hem on our lab repor. N.B. This lab is a formal one so even hough sill based on eam work each suden will have o urn in an individual lab repor. Quiz on he ne page) 4
PHYS 154 Universi Phsics Laboraor Pre-Lab Spring 18 C. Quiz 5 Name: Insrucions: The firs hree quesions on he quiz require ha ou observe he moion of a projecile in a simulaion accessible using he course homepage. So eiher go o he homepage and click on he [Animaions] buon in he upper lef corner or go direcl o people.morrisville.edu/~freamamv/secondar/animaions/cm.hml. Click [CM] on he lef-hand-side lis or scroll down o he animaion dubbed CM showing a red projecile read o launch. The red arrows represen he vecor veloci and is componens. The insrucions for using he buons are on he lef. Don worr abou he echnicaliies abou he air resisance (drag) jus hink abou i in qualiaive erms as a force alwas acing agains he direcion of moion. Q1. [] Make sure ha he [Drag] bo is unchecked. Click he [Sar] buon and observe he horizonal componen v of he veloci. Which of he following is an eplanaion for he behavior of his componen? a) In he firs half he weigh deceleraes he projecile so v ges smaller and smaller. b) In he second half he weigh acceleraes he projecile so v ges larger and larger. c) As i moves he projecile loses momenum so v ges smaller and smaller hroughou. d) In he firs half he force delivered b he launcher cancels he weigh so v sas consan. e) There is no horizonal force acing on he projecile so v sas consan hroughou. Q. [] Make sure ha he [Drag] bo is unchecked. Click he [Sar] buon and le he projecile complee a rajecor. Then check he [Drag] bo. Click [Rese] o bring he projecile a sar and launch i again. [Rese] and repea he launch wih he drag srenghened b increasing he k-consan o. Which of he following is an eplanaion for he shapes of he hree observed rajecories? a) All rajecories are lef-righ smmeric because he forces ac in he same wa hroughou. b) Onl he no-drag rajecor is lef-righ smmeric because here is no horizonal force on he projecile. c) The sronger is he drag he aller he rajecories because air resisance compresses hem horizonall. d) The sronger is he drag he shorer and asmmeric he rajecories because air resisance decreases he - componen of he veloci. e) Boh (b) and (d) are rue. Q3. [] Uncheck he [Drag] bo. Click [Rescale] o clear he screen. Repea he projecile launch for he following iniial angles: θ =5º 35º 45º 55º 65º (make sure ha ever ime ou change he angle in he bo ou click in anoher bo for he new value o ake in he vecors should redirec o follow he new angle). Which of he following is rue abou he observed rajecories? a) The larger he iniial angle he shorer he rajecories. b) The same horizonal range can be reached for wo iniial angles: one lower and he oher larger han 45º. c) The maimum range is reached when θ = 45º. d) The larger he iniial angle he longer is he ime spen b he projecile in he air. e) Answers (b) (c) and (d) are all rue. Q4. [] Wha is he main goal of our ne lab eperimen? a) To measure he muzzle speed of a projecile. b) To measure he disance raveled b a projecile fired and landing a he same verical level. c) To measure he disance raveled b a projecile fired horizonall from a desk and landing on he floor. d) To verif ha he predicion made b he wo-dimensional kinemaic model for projecile moion is confirmed b he moion of an acual projecile. e) None of he above. Q5. [] Sa ha in PART of our ne lab our launcher is 1.1 meers above he floor. Your projecile is launched wih an iniial speed of 3.4 m/s a an angle of 3º. Wha is he horizonal disance ha ou should epec our projecile o ravel before i ouches he floor? (Hin: read and follow he Technical Commens in he pre-lab.) a) Abou meers b) Abou 1.8 meers c) Abou.7 meers d) Abou 1 meer e) None of he above. 5