Computer models of motion: Iterative calculations

Transcription

1 Computer models o moton: Iteratve calculatons OBJECTIVES In ths actvty you wll learn how to: Create 3D box objects Update the poston o an object teratvely (repeatedly) to anmate ts moton Update the momentum and poston o an object teratvely (repeatedly) to predct ts moton TIME You should plan to nsh ths actvty n 50 mnutes or less. COMPUTER PROGRAM ORGANIZATION A computer program conssts o a sequence o nstructons. The computer carres out the nstructons one by one, n the order n whch they appear, and stops when t reaches the end. Each nstructon must be entered exactly correctly (as t were an nstructon to your calculator). I the computer encounters an error n an nstructon (such as a typng error), t wll stop runnng and prnt a red error message. A typcal program has our sectons: 1. Setup statements 2. Dentons o constants ( needed) 3. Creaton o objects and speccaton o ntal condtons 4. Calculatons to predct moton or move objects (done repettvely n a loop ) I. Setup statements Usng IDLE or VPython, create a new le and save t to your own student drve. Make sure to add ".py" to the le name. Enter the ollowng two statements n the IDLE edtor wndow: rom uture mport dvson rom vsual mport * Remember that every VPython program begns wth these setup statements. The rst statement (rom space underscore underscore uture underscore underscore space mport space *) tells the Python language to treat 1/2 as 0.5. Wthout the rst statement, the Python programmng language does nteger dvson wth truncaton and 1/2 s zero! The second statement tells the program to use the 3D module (called vsual ). 2. Constants Followng the setup secton o the program you would dene physcs constants. We ll talk about ths n later projects. 3a. Creatng objects Create a box object to represent the track: 1

2 track = box(pos=vector(0,-.05, 0), sze=(2.0, 0.05,.10), color=color.whte) Run the program by pressng F5. Arrange your wndows so the Python Shell wndow s always vsble. Kll the program by closng the graphc dsplay wndow. Now create a second box object, named "cart", wth some color other than whte. Gve ths object a poston (pos) o (0,0,0) and a sze o (0.1, 0.04, 0.06). Run the program by pressng F5. Zoom (both mouse buttons down) and rotate (rght mouse button down) to examne the scene. The cart should be loatng just above the track. Is t? Reposton the cart so ts let end s algned wth the let end o the track. To do ths you wll have to answer the ollowng questons: Where s the "pos" o a box object? The let end? The rght end? The center? Do the numbers n the "sze" o a box reer to the total length, or the dstance rom the center to one edge? You can answer these by expermentaton, or by lookng n the onlne reerence manual (Help menu, choose Vsual, clck on Reerence Manual.) 3b. Intal condtons Any object that moves needs two vector quanttes declared beore the loop begns: 1. ntal poston; and 2. ntal momentum. You ve already gven the cart an ntal poston at the let end o the track. Now you need to gve t an ntal momentum. I you push the cart wth your hand, the ntal momentum s the momentum o the cart just ater t leaves your hand. Snce the denton o momentum at speeds much less than the speed o lght s p mv, we need to tell the computer the cart s mass and the cart s ntal velocty. Below the exstng lnes o code, type the ollowng new lnes: mcart = 0.80 pcart = mcart*vector(0.5, 0, 0) prnt ('cart momentum =', pcart) We have made up a new varable named mcart The symbol mcart now stands or the value 0.80 (a scalar), whch represents the mass o the cart n klograms. We have also created a new vector varable pcart, whch s the momentum o the cart. We assgned t the ntal value o ( 0.80 kg) 0.5, 0, 0 m/s. Run the program. Look at the Python Shell wndow. Is the correct value o the vector pcart prnted there? From what s prnted, how can you tell t s a vector? Note: There are no bult n physcs attrbutes p or m or objects lke there are bult-n geometrcal attrbutes pos or radus. However, Python allows us to create new attrbutes or objects. We could have called the momentum cart.p, or the mass cart.m, nstead o pcart or mcart. It can be helpul to create attrbutes lke mass or momentum assocated wth objects so we can easly tell apart the masses and momenta o derent objects n a complex program. 2

3 3b. Tme step and total elapsed tme To make the cart move we wll use the poston update equaton r r vt repeatedly n a loop. We need to dene a varable deltat to stand or the tme step t, and a varable t to stand or the total tme elapsed snce the moton started. Here we wll use the value t = 0.01 s. Type the ollowng new lnes at the end o your program: deltat = 0.01 t = 0 Ths completes the rst part o the program, whch tells the computer to: a. Create numercal values or constants we mght need (none were needed ths tme) b. Create 3D objects c. Gve them ntal postons and momenta 4. Repeated calculatons: Loops In a computer program a sequence o nstructons that are to be repeated s called a loop. The knd o loop we wll use n VPython starts wth a "whle" statement. Instructons nsde the loop are ndented. IDLE wll ndent automatcally ater you type a colon. To wrte a smple loop, type the ollowng new lnes at the end o your program. Be sure to type a colon (:) at the end o the whle statement. Make sure the ndentng s correct, as shown below, then run: whle t < 0.2: prnt 'the tme s now', t t = t + deltat prnt 'ater the loop' The statement: t = t + deltat may look lke a mathematcal error. However, n a program, the "=" sgn has a derent meanng than n a mathematcal equaton. The rght hand sde o the statement tells Python to read up the old value o t, and add the value o deltat to t. The let hand sde o the statement tells Python to store ths new value nto the varable t. Run the program. Look at the Python Shell wndow. Look at the prnted output n the Shell wndow. Answer the ollowng questons: What makes the loop stop? Why s the rst prnted tme 0? Why s the last tme 0.19 and not 0.2? How can you get the program to prnt values rom 0 through 0.3? (Try t.) 4a. Constant momentum moton Consder a cart movng wth constant momentum. Somebody or somethng gave the cart some ntal momentum. We re not concerned here wth how t got that ntal momentum. We ll predct how the cart wll move n the uture, ater t acqured ts ntal momentum. 3

4 You wll use your teratve calculatonal loop. Each tme the program runs through ths loop, t wll do two thngs: 1. Use the cart s current momentum to calculate the cart s new poston 2. Increment the cumulatve tme t by deltat You know that the new poston o an object ater a tme nterval r r v t avg t s gven by where r s the nal poston o the object, and r s ts ntal poston. I the tme nterval t.s very short, so the velocty doesn t change very much, we can use the ntal or nal velocty to approxmate the average velocty. Snce at low speed p mv, or v p / m, we can wrte r r ( p / m) t We wll use ths equaton to ncrement the poston o the cart n the program. Frst, we must translate t so VPython can understand t. Delete or comment out the lne nsde your loop that prnts the value o t. On the ndented lne ater the whle statement, and beore the statement updatng t, type the ollowng: cart.pos = cart.pos + (pcart/mcart)*deltat Notce how ths statement corresponds to the algebrac equaton: r r ( p / m) t cart.pos = cart.pos + (pcart/mcart)*deltat Fnal poston Intal poston Velocty Tme step Thnk about the stuaton and answer the ollowng queston: What wll the elapsed tme t be ater movng two meters? Change the whle statement so the program runs just long enough or the cart to travel 2 meters. Now, run the program. What do you see? Slowng down the anmaton When you run the program, you should see the cart at ts nal pont. The program s executed so rapdly that the entre moton occurs aster than we can see, because a "vrtual tme" n the program elapses much aster than real tme does. We can slow down the anmaton rate by addng a rate statement. Add the ollowng lne nsde your loop (ndented): 4

5 rate(100) Every tme the computer executes the loop, when t reads rate(100), t pauses long enough to ensure the loop wll take 1/100 th o a second. Thereore, the computer wll only execute the loop 100 tmes per second. Now, run the program. You should see the cart travel to the rght at a constant velocty, endng up 2 meters rom ts startng locaton. Note: The cart gong beyond the edge o the track sn t a good smulaton o what really happens, but t s what we told the computer to do. There are no bult-n physcal behavors, lke gravtatonal orce, n VPython. Rght now, all we ve done s tell the program to make the cart move n a straght lne. I we wanted the cart to all o the edge, we would have to enter statements nto the program to tell the computer how to do ths. Answer the ollowng questons: 1. Whch statement n your program represents the poston update ormula? 2. What would you have to change n your program to make the cart start at the rght end o the track and move to the let? Do ths. When you have succeeded, compare your program to that o another group. 4b. 2D moton In a computer program you can model behavor that would be dcult to observe n the real world. Do the ollowng: Change the ntal momentum o the ancart so that t ncludes a +y component smlar n magntude to the x component o the momentum. What happens? Explan ths, then compare your explanaton to that o a neghborng group. Usng VPython on your own VPython s ree. You can download VPython rom and nstall t on your own computer. VPython s also avalable n the campus publc clusters n the Math, Statstcs, and Physcs secton o the Novell Applcaton Launcher; doubleclck "IDLE or VPython". In the text edtor (IDLE), on the Help menu you can choose Vsual, then Reerence manual, or choose Python Docs to obtan detaled normaton on the Python programmng language upon whch VPython s based. We wll use only a small subset o Python s extensve capabltes. 5