Lab 5 Projecile Moion 71 Name Dae Parners OVEVIEW LAB 5 - POJECTILE MOTION We learn in our sudy of kinemaics ha wo-dimensional moion is a sraighforward exension of one-dimensional moion. Projecile moion under he influence of graviy is a subjec wih which we are all familiar. We learn o shoo baskeballs in an arc o swish hrough he baske or o bounce off he backboard. We learn how o lob in volleyball and ennis. These are examples of projecile moion. The force of graviy acs in he verical direcion, and air resisance acs in boh he horizonal and verical direcions, bu we ofen neglec air resisance for small objecs. In his experimen we will explore how he moion depends upon he body s iniial velociy and elevaion angle. Consider a body wih an iniial speed v a angle α wih respec o he horizonal axis. We analyze he body s moion in wo independen coordinaes x (horizonal) and y (verical). [As we are free o choose his origin of our coordinae sysem, we choose x = and y = when = so as o simplify our calculaions.] Figure 1 hen shows ha he componens of he velociy vecor along he x and y axes are respecively: v x = v cosα and v y = v sinα (1) If we neglec air resisance, he only force affecing he moion of he objec is graviy, which near he Earh s surface acs purely in he verical direcion ( F ˆ g = mg = mgy ). There is no force a all in he horizonal direcion. Since here is no horizonally applied force, here is no acceleraion in he horizonal direcion; hence he x direcion of he velociy will remain unchanged forever. The horizonal posiion of he body is hen described by he expression for consan velociy: x ) = v x ( v cosα ) () ( = The force in he y (verical) direcion is graviaion ( Fy = mg ). Since F = ma, ay Inegraing his wih respec o ime yields he verical componen of he velociy: v y ( V V y α = g. ) = v y g = v sinα g. (3) The verical componen of he posiion can be obained by inegraing Equaion (3) wih respec o ime, yielding he resul: V x Figure 1 1 1 y( ) = v y g = ( v sinα ) g. (4) PHYS 14W, Spring 7
7 Lab 5 Projecile Moion We can use hese equaions o deermine he acual rajecory of he body in erms of he x and y variables, wih no explici reference o he ime. The equaion has a parabolic form y C x C x = 1 (5) where C1 = an ( α) and C = g / v cos ( α). We are ineresed in he range, he horizonal disance ha he projecile ravels on level ground. We can ge he range from Equaion (5) by noing ha y sars a zero when x = and is once again zero when x =. We ge v sin( ) α = (6) g Alhough we won measure hem in his experimen, we can similarly show ha he ime of fligh (T ) is given by T v g and he maximum heigh (reached when = T / ) is given by h α = sin ( ) (7) v sin ( α) g INVESTIGATION 1: MEASUEMENT OF ANGE = (8) You will need he following maerials: projecile moion apparaus landing pad opical bench digial phoogae imer meer sick masking ape plasic ruler 3 m ape measure level spark (pressure-sensiive) paper Figure 1. Before saring he measuremens, familiarize yourself wih he apparaus. The cenral feaure of he seup is a spring-loaded gun mouned on a plae marked in degrees, as shown in Figure. The plae can be roaed around a horizonal axis o se he iniial PHYS 14W, Spring 7
Lab 5 Projecile Moion 73 direcion of moion: a desired angle can be se by loosening he knurled screw aached o he plae, aligning he required angle mark wih he horizonal mark on he sand and hen ighening he screw. The iniial speed of he projecile (a seel ball) can be se o hree differen values by drawing back he spring-loaded firing pin unil i is locked in one of he hree prese posiions. Noice he cable-release rigger and make sure ha is plunger is fully wihdrawn (his is achieved by pressing on he small ring near he end of he rigger cable), hen ry drawing back he firing pin and lisen o he hree disinc clicks corresponding o he hree posiions. Unforunaely, someimes he cablerelease becomes disconneced, somehing you wan o avoid, because i is difficul o reconnec i! Press he rigger s plunger and he pin will fire. Now you can pracice launching he projecile: ry differen angles and differen velociies o ge a feeling of which angular range will land he projecile ono he able. When doing your measuremens, record he landing spo of he ball on he able by means of a pressure sensiive paper aped o he board. Do no ape he paper in posiion unil you are ready o sar your measuremens, because i is expensive and we can afford o wase any.. Now you will wan o verify ha he apparaus is level and ha he landing able is a he same heigh as he pivo poin (which is where we believe he ball also leaves he spring). Firs, place he level on he back of he gun apparaus and, if needed, use he wo screws holding up he apparaus o level he gun apparaus. Use a meer sick and he level o make sure ha he able op surface is a he same heigh as he pivo poin of he angle mark (he iming apparaus can easily be pulled ou o see where he pivo poin acually is) and ha he able is level. Ask your TA for assisance if his procedure is no clear. Aciviy 1-1: Deermining he Iniial Speed 1. As he final preparaion, we wan o measure he iniial speed of he projecile. As i leaves he gun, he projecile crosses he ligh beam of wo ligh bulbs (acually ligh emiing diodes or LED s), shining ono wo phoocells, which are conneced o a imer uni. When in pulse mode, he imer will sar couning when he firs ligh beam is inerruped, and i will keep couning unil he second beam is obscured. Make sure he imer is se o.1 ms resoluion and se he angle o. Wihdraw he firing pin o one of he hree posiions and place he ball ino he gun. If he imer is running, press he ESET buon. Launch he ball and use a Syrofoam cup o cach he ball. The imer will record he ime in seconds i ook he projecile o go from he firs o he second ligh. Knowing he disance d beween he wo (i is marked on he apparaus), you can deermine he average speed of he ball as i ravels beween he lighs. Take one half of he leas significan digi in d as, your esimae of your uncerainy in d. d d : d : PHYS 14W, Spring 7
74 Lab 5 Projecile Moion Quesion 1-1: Can you hink of a beer way o measure he iniial speed? Discuss among your group why we use his procedure ( angle) o deermine he iniial speed. Wrie your conclusions here. Measure he ime ( ) for a leas four rials for each click seing. Ener your daa ino Table 1-1. Use hese daa and he posiion beween he phoocells o deermine he speed. For each seing calculae he average ime, ( ) and he sandard deviaion,. Le your uncerainy in ( ), be he larger of he sandard error in he mean or one-half of he leas significan digi (.5 ms). Calculae he iniial speed, v = d, and is uncerainy, v o, based on your esimaes of d and Click Posiion 1 3 (ms) Trial 1 Trial Trial 3 Trial 4. Table 1-1 Iniial Speed (ms) Aciviy 1-: ange as a Funcion of Iniial Angle (ms) (ms) v (m/s) v (m/s) For his aciviy you will choose one of he hree values of he launch speed and execue a se of launches for a leas five differen values of he launch angle, say 15, 3, 45, 6, and 75. We sugges ha you ake daa wih highes iniial speed. Quesion 1-: Wrie down wo reasons why i migh be bes o use he highes speed o ake hese daa. Predicion 1-1: Fix your coordinae sysem such ha he x and y coordinaes are boh zero when he ball passes hrough he pivo and adjus he op of he landing board o ha level ( y = ). The ball hen sars a y = and reaches y = again when landing on he PHYS 14W, Spring 7
Lab 5 Projecile Moion 75 board. The range will simply be he x coordinae where i his. Draw your predicions for he rajecory on he following graph for each of your chosen angles 3 Projecile Moion Trajecory y [cm] 5 15 1 5 1 3 4 5 6 7 8 9 1 11 1 x [cm] Predicion 1-: Draw your heoreical predicions for he range as a funcion of he iniial angle α on he following graph. 1 1 ange [cm] 8 6 4 3 6 9 a [degrees] Quesion 1-3: If you waned o plo he range versus a quaniy ha depends on angle (ha is, you are going o vary he angle) and ha would heoreically resul in a sraigh line for all five iniial angles, wha would you choose for he independen funcion variable (funcion variable ploed along he horizonal axis)? Explain. PHYS 14W, Spring 7
76 Lab 5 Projecile Moion 1. emove he phoogae so ha you can more easily measure he disances. Turn off he phoogae s power.. For each launch, he range of he projecile will be recorded by a do on he pressuresensiive paper. You will noice ha even if you keep he iniial speed and inclinaion fixed, he landing posiions will be spread over several millimeers. The acual range you will use for ploing your daa will be he average of he measured poins. To avoid confusions due o spurious marks on he paper caused by bounces of he projecile, i is advisable, firs, o label each mark as he ball makes i and, second, o perform he measuremens wih he longes ranges firs. You migh, for example, circle each mark o disinguish i from he newer ones. Do no ake daa unil insruced o do so! Quesion 1-4: Can you deermine from he formula derived earlier which angle corresponds o he longes range? Explain. Tha should be your firs angle. 3. ecord your iniial speed: v 4. Now ake your daa for his same v and inser your values ino he following able: Trial 1 3 4 5 Table 1- ange (in cm) for each projecile launch Angle 15 3 45 6 75 PHYS 14W, Spring 7
Lab 5 Projecile Moion 77 5. For each angle, calculae he average range,, and he sandard deviaion,. Le your uncerainy in,, be he larger of he sandard error in he mean or one half of he leas significan digi. Quesion 1-5: For which angle is he range he larges? Does his agree wih your predicion? Explain. Quesion 1-6: Does i appear from your daa ha he ranges for any of he angles are he same (wihin he probable errors)? If so, for which ses of daa are his rue? Discuss. 6. Now ha you have your daa, compare hem wih he heoreical predicions. You wan o plo your daa using he funcion you decided in Quesion 1-3 ha produces a sraigh line. For each angular seing, use he expression you derived earlier o compue he expeced projecile range as some funcion of angle. Produce he line for he heoreical predicions (see = [ v sin( α)]/ g ) and use your measured values of v and he known value of g. Calculae for each angle and pu he resuls and your daa ino Excel. Then you can connec he daa poins wih a line, bu choose in Excel o no display he calculaed daa poins. Show only he heoreical line and he experimenal daa versus a suiable parameer ha depends on he iniial angle α (see Quesion 1-3) using Excel. The resul should be a linear plo. 7. Prin ou and include he graph wih your repor. 8. By hand, add error bars o he measured daa on your graph. These should be shor lines of lengh ha exend above and below he daa poin. Ofen hese lines have T like ends. PHYS 14W, Spring 7
78 Lab 5 Projecile Moion Quesion 1-7: How well do your measuremens agree wih your predicions? Are he deviaions consisen o wihin he accuracy you expec of your measuremens? Explain why or why no. Discuss possible sources of sysemaic errors ha may be presen. Aciviy 1-3: ange as a Funcion of Iniial Speed 1. Now you wan o choose one suiable angle and measure he range for each of he hree iniial speeds ha you deermined earlier. Wrie down your chosen angle: Launch angle Quesion 1-8: Wha crieria did you use for choosing your launch angle? Would i beer o use a shallow or high angle? Does i maer for any reason?. Take a leas four readings for each of he hree iniial speeds. Inser your resuls ino Table 1-3. PHYS 14W, Spring 7
Lab 5 Projecile Moion 79 Trial 1 3 4 5 Table 1-3 ange (in cm) as a Funcion of Iniial Speed Iniial Velociy 1: : 3: Quesion 1-9: I would be insrucive o plo your resuls again on a graph ha produces a linear line. Look a he equaion for range and deermine wha you should plo he range versus. Sae your choice and give an explanaion: 3. Plo your experimenal values for he range on a graph versus he parameer you chose in he previous quesion and compare hem as before wih a line drawn by he heoreical values prediced by he expression you derived. 4. Prin ou your graph, add error bars and aach he graph o your repor. Quesion 1-1: Do your daa agree reasonably well wih your heoreical predicions? How many of your daa poins have error bars ha overlap your heoreical calculaion? Is his wha you expec? Do you believe your errors are saisical or sysemaic? Explain. PHYS 14W, Spring 7
8 Lab 5 Projecile Moion PHYS 14W, Spring 7