LAB 4: PROJECTILE MOTION

Transcription

1 55 Name Dae Parners LAB 4: POJECTILE MOTION A famous illusraion from Newon s Principia showing he relaionship beween projecile moion and orbial moion OVEVIEW We learned 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: Figure 1. Velociy vecor v x = v cosα and v y = v sinα (1)

2 56 Lab 4 Projecile Moion 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 componen 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 ( = g. ) = v y g = v sinα g. (3) The verical componen of he posiion can be obained by inegraing Eq. (3) wih respec o ime, yielding he resul: 1 1 y( ) = v y g = ( v sinα ) g. (4) 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 heigh h and he range, he horizonal disance ha he projecile ravels on level ground. We can ge he range from Eq. (5) by noing ha y sars a zero when x = and is once again zero when x =. We obain (see your exbook for deails) 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 v T = sin ( α) (7) g and he maximum heigh h = y max is he value of y when he ime is T/. I also occurs when v y =. The value of h is given by v sin ( α) h = (8) g We show he rajecory in Figure wih he iniial angle α, range, and maximum heigh h indicaed.

3 Lab 4 Projecile Moion 57 Figure. Projecile rajecory We have prepared an Excel file o help you immensely in his experimen. Open Excel, find he TeachingLabs direcory, open 3, and open he file L4.Projecile Moion. Noe a he boom here are hree abs (called shees): Table 1-1, Table 1-, and Table 1-3 for he hree differen measuremens you will be making. You will be enering your daa ino he yellow background areas. Do nohing wih he oher cells, because Excel will do he calculaions for you. INVESTIGATION 1: MEASUEMENT OF ANGE You will need he following maerials: projecile moion apparaus landing pad opical bench digial phoogae imer meer sick masking ape level 3 m ape measure spark (pressure-sensiive) paper Figure 3. Projecile moion apparaus and landing pad. 1. Before doing anyhing, lisen while he TA goes hrough a descripion of he apparaus and how o level i. I is crucial for he apparaus o be level or you will no obain good resuls. The wooden able is heavy and can easily slip during he measuremens.

4 58 Lab 4 Projecile Moion. Afer he brief TA discussion, 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 Fig. 3. The plae can be roaed around a horizonal axis o se he iniial 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. 3. 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 cable-release becomes disconneced, somehing you wan o avoid, because i is difficul o reconnec i! Press he rigger s plunger and he pin will fire. 4. 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. 5. 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. IT IS CUCIAL THAT YOU HAVE DONE THIS COECTLY! ACTIVITY 1-1: DETEMINING THE INITIAL 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 LEDs), 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 launcher angle o. This will give you he bes possible ime resoluion.. 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 he Syrofoam cup o cach he ball. The imer will record he ime in seconds i ook he projecile

5 Lab 4 Projecile Moion 59 o go from he firs o he second ligh. If you know he disance d beween he wo (i is marked on he apparaus and should be mm), you can deermine he average speed of he ball as i ravels beween he lighs. Wihou more knowledge, we assume he uncerainy σ in d is one half of he leas significan digi in d or.5 mm. d d : ±.5 mm Quesion 1-1: Can you hink of a differen way o measure he iniial speed? Describe a leas one oher mehod. Discuss among your group why we use he procedure suggesed ( angle) o deermine he iniial speed. Wrie your conclusions here. 3. You should have he Excel file L4.Projecile Moion open (see paragraph above) and make sure you are looking a Table 1-1 (if no here, click on he proper shee near he boom of he screen). Noe ha even hough he imer is se o.1 ms, his refers o is resoluion. The imer sill displays in seconds. I is imporan o work ogeher. Measure he ime ( ) for a leas four rials for each click seing. One suden should ener he daa ino he Excel file while he oher sudens ake he daa. Take urns during he lab doing each. Alhough your primary daa record is in Excel, i is wise o also wrie your daa ino Table 1-1 shown here in he journal in case somehing happens o your compuer file. Do no calculae he quaniies in Table 1-1 by hand Excel does his for you. 4. The Excel file will perform he calculaions for you, bu i is absoluely necessary ha you undersand wha is being calculaed. You can click on a given cell in Excel and look above he column headings o see he funcion being calculaed. For example in cell F5, you will see he following: =AVEAGE(B5:E5)*1 This means ha we are finding he average ime in seconds beween cells B5 and E5 and hen convering he ime o milliseconds (ms) by muliplying by 1. You should click on each cell in a row o undersand wha is being calculaed. For each seing he Excel file calculaes he average ime ( avg =, Column F), he sandard deviaion σ (sig, Column G), he sandard error in he mean ( σ ) (san error mean, Column H), and one-half of he leas significan digi in he imer (.5 ms, Column I). Le your uncerainy in ( σ ), be he larger of he sandard error in he mean or one-half of he leas significan digi (.5 ms) and ha is in he nex Column J headed by larges sigbar. In Column M we calculae he iniial speed, v = d /. In order o calculae he uncerainy in he velociy we need o use he propagaion of

6 6 Lab 4 Projecile Moion Posiion Click 1 3 errors, because we have uncerainies in boh disance d and ime. Therefore in Columns K and L we find he fracional errors in he ime and disance, respecively. These values are used o deermine he uncerainy in velociy given in Column N. Finally in Column O we find he fracional uncerainy in he velociy, which we will use laer o deermine he uncerainy in a calculaion of he range.. Trial 1 Trial (s) Trial 3 Trial 4 Table 1-1 Iniial Speed (ms) σ (ms) sand error mean σ (ms) sig digi (ms) larges sigbar (ms) frac error frac error d vel v (m/s) uncer vel σ v (m/s) frac uncer vel 5. Prin ou one copy of Table 1-1 from Excel for your group repor. Quesion 1-: Look a he funcion being calculaed in cell N6 of he Excel file. Wha is being calculaed? Explain he equaion being used. ACTIVITY 1-: ANGE AS A FUNCTION OF INITIAL ANGLE 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: 15, 3, 45, 6, and 75. We sugges ha you ake range daa wih he highes iniial speed.

7 Lab 4 Projecile Moion 61 Quesion 1-3: Wrie down wo reasons why i migh be bes o use he highes speed o ake hese daa. Predicion 1-1: Consider a coordinae sysem where he x and y coordinaes are boh zero when he ball passes hrough he pivo and lands on he op of he landing board a ha same level ( y = ). The ball hen sars a y = and reaches y = again when landing on he board. The range will simply be he x coordinae where i his. Draw your qualiaive predicions for he rajecory on he following graph for 15, 3, 45, 6, 75. (These will be lines of moion.) Do his before coming o lab. [Qualiaive means do no use acual numbers, bu make he rajecories correc relaive o each oher: show which goes highes, which goes furher, which have he same rajecory, ec.] Predicion 1-: Draw your heoreical predicions for he range as a funcion of he iniial angle α on he following graph. (These will be poins.) You did a similar calculaion in he prelab. You may include your daa here from your prelab calculaion. Do his before coming o lab.

8 6 Lab 4 Projecile Moion ange (m) Iniial Angle a (degrees) 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! *Ask your TA for permission o proceed*. Quesion 1-4: Can you deermine from he formula given earlier (Eq. (6)) which angle corresponds o he longes range? Explain. Tha should be your firs angle. 3. ecord your iniial speed: v :. Ener his speed in Table 1- of your Excel file. You may have o click on he correc shee in Excel a he boom of he screen. 4. Now ake your daa for his same v and inser your values ino Table 1- of he Excel file. Your TA may have some suggesions as o how o mos easily measure he individual ranges. I is easies o measure in cenimeers, and we will conver o meers in he Excel file. You may wan o have one member of your group also include he range measuremens in Table 1- of his journal for securiy reasons (wha if somehing

9 Lab 4 Projecile Moion 63 happens o your Excel file?), bu i is no required. emember o work ogeher. One suden can inpu he daa ino Excel, while he oher sudens perform he measuremens. Move quickly along o ake hese daa. 5. You do no need o do he calculaions of, σ, and σ shown in Table 1-. Excel will do his for you. We show you Table 1- here for purposes of knowing wha is being done in your lab. Table 1- ange for each projecile launch Trial ange 1 (cm) ange (cm) ange 3 (cm) ange 4 (cm) ange 5 (cm) Calculaions (m) σ (m) San error mean σ (m) heory (m) Angle The calculaions below are done for you in Excel Quesion 1-5: Think abou your measuremen of he range. How accurae do you hink ha measuremen is? A he very mos i can be no beer han you can read he measuremen device (meer sick, ape measure, whaever...) Wrie below wha you hink his limiing measuremen value is and ener i ino Table 1- and ino row 14 of he Excel file. Measuremen limi: Explain how you deermined his value: 6. For each angle, Excel calculaes he average range,, he sandard deviaion, σ, and he sandard error in he mean σ. Le your uncerainy in be he larger of he

10 64 Lab 4 Projecile Moion sandard error in he mean or he value of your measuremen limi from Quesion 1-5. The uncerainy in is finally given in ow 15. Quesion 1-6: Does i appear from your daa ha he ranges for any of he angles are abou he same? If so, for which ses of daa are his rue? Discuss. Do you expec any angles o have he same range? Explain. 7. Look carefully a Table 1- in your Excel file o see wha is being calculaed. For example, he heoreical range is calculaed in Column J depending on he angles shown in Column H and he velociy given in Cell I15. The average experimenal range is deermined in Column K. The graph shows he heoreical range ploed versus sin (α) and is compared wih your experimenal measuremens. The heory calculaion depends on he velociy measuremen you made in he previous aciviy. Quesion 1-7: heory is being calculaed in column J. Wrie down he equaion used in erms of parameers (no cell numbers) and explain where he equaion comes from. Quesion 1-8: Explain wha is being calculaed in cell D13 of Table 1- in Excel. Wrie down he equaion used in his cell below. Wha equaion on page D-6 of Appendix D are we deermining? 8. Prin ou Table 1- in Excel and include he graph wih your repor. 9. Try o add error bars by hand o he measured daa on your graph. These should be shor lines of your maximum uncerainy in ha exend above and below he daa poin. Ofen hese lines have T like ends. See he nex quesion if you have a problem.

11 Lab 4 Projecile Moion 65 Quesion 1-9: Are your uncerainies in smaller han he size of your daa poins? If so, is ha good? Explain. Quesion 1-1: How well do your measuremens agree wih your predicions? Are he deviaions consisen wih he accuracy you expec of your measuremens? Explain why or why no. Discuss possible sources of sysemaic errors ha may be presen. ACTIVITY 1-: ANGE AS A FUNCTION OF INITIAL 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 1-11: 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 five readings for each of he hree iniial speeds. Inser your resuls ino Table 1-3 of he Excel file. Ener he daa in cenimeers, and we will conver i o meers in Excel. As a precauion agains losing your daa, perhaps one suden should wrie he daa in Table 1-3 of his journal, bu i is no required. 3. The Excel file will calculae he average range daa and uncerainies as before. This informaion is here so you will know wha we are doing in he lab. 4. Excel has ploed your range daa versus v. The plo also indicaes he heoreical range calculaed using he velociies you deermined in Aciviy 1-1 and show in Table 1-1 of he Excel file. Because here are uncerainies in he velociy measuremens (see Columns N and O of Table 1-1), here are also uncerainies in he calculaion of our heoreical ranges. The error bars on he heory calculaions in

12 66 Lab 4 Projecile Moion Table 1-3 are reflecive of he velociy uncerainies. We do he range heoreical calculaions in ow 13 and deermine he uncerainy in ow 14. Table 1-3 ange as a Funcion of Iniial Speed Trial Iniial Velociy 1: : 3: ange 1 (cm) ange (cm) ange 3 (cm) ange 4 (cm) ange 5 (cm) (m) σ (m) San error meanσ (m) Measuremen limi (m) heory (m) San error in heory (m) v (m/s) Quesion 1-1: Examine Cell D14 in Table 1-3 of he Excel file. Wrie down he equaion in he cell below. Wha is being calculaed? Explain each erm in he equaion. 5. Prin ou Table 1-3 from Excel and your graph. Add error bars o your range daa (if larger han he daa poins), and aach he graph o your repor. Quesion 1-13: Do your experimenal daa agree reasonably well wih your heoreical predicions? How many of your daa poins overlap your heoreical calculaion? Is his wha you expec? Do you believe your errors are saisical or sysemaic? Explain.