Projects Structure o Project Reports: 1 Introducton. Brely summarze the nature o the physcal system. Theory. Descrbe equatons selected or the project. Dscuss relevance and lmtatons o the equatons. 3 Method. Descrbe brely the algorthm and how t s mplemented n the program. 4 Vercaton o a program. Conrm that your program s not ncorrect by consderng specal cases and by gvng at least one comparson to a hand calculaton or known result. 5 Results. Show the results n graphcal or tabular orm. Addtonal runs can be ncluded n an appendx. Dscuss results. 6 Analyss. Summarze your results and explan them n smple physcal terms whenever possble. 7 Crtque. Summarze the mportant concepts or whch you ganed a better understandng and dscuss the numercal or computer technques you learned. Make specc comments on the assgnment and your suggestons or mprovements or alternatves. 8 Appendx. Gve a typcal lstng o your program. The program should nclude your name and date, and be sel-explanatory (comments, structure) as possble. The report should be as concse as possble. 1
Project 1: A smple projectle moton (no ar resstance) (Due on Frday, February 1, 1 by 13:3) A Brtsh navy shp s about 1 nautcal mle rom a ort deendng Tortuga. (In 17th century Tortuga was one o the largest prate strongholds). The ortress s located 5. meters above the sea level, wth the walls as hgh as 6. meters. There s an armory house located 1. meters beyond the ort walls. The captan o the navy shp knows that a drect ht o the armory house (stued wth barrels o rum) may orce prates to surrender. The shp can re cannons at the muzzle speed o v= m/s. Let's suppose that the captan can dsregard ar resstance n the problem (We wll consder the eect o the ar resstance later). 1 At what angle rom the horzontal must the cannons be red to ht the armory? (Use one o methods or solvng non-lnear equatons) Is t mportant to take nto account that the shp s actually movng toward the ort wth the speed about knots? (1 knot = 1 nautcal mle/hour = 1.85 km/h). The armory s dmensons are 8.m*8.m*3.m. 3 What would you do you were the captan o the navy shp? 4 Bonus: Evaluate the mportance o the eect o ar resstance (Back o the Envelope Physcs). Equatons: In the smplest case (wth no ar resstance) the D moton o a projectle s descrbe by a system o equatons x = x + ( v cos + v ) t θ shp gt y = y + ( v snθ ) t Elmnatng the tme rom the equatons gves x x g x x y sn y + v θ = v cosθ + v cos shp v θ + vshp Solvng ths non-lnear equaton or the angle θ would gve the rght shootng angle to ht a target. In the case y = y and v = a very smple analytc soluton can be ound n most textbook θ shp 1 g( x x ) arcsn v = Ths smple case can be used to test numercal solutons when y = y and v =. shp
Project : Projectle moton wth ar resstance (Due on Wednesday, March 5, 1 by 1:) Wrte a program that smulates the projectle moton n the (x,y) plane wth allowng or ar resstance, varyng ar densty and wnd. The ampltude o ar resstance orce on an object movng wth speed v can be approxmated by F drag =-.5Cρ Av, where ρ stands or ar densty (ρ =1.5 kg/m 3 at sea level), and A s the cross secton. The drag coecent C depends on an object shape and or many objects t can be approxmated by a value wthn.5 -.5. Use Runge-Kutta method as a prmary method or solvng a system o derental equatons. Applcaton: Study the trajectory o shells o one o the largest cannons "Parskanone" used durng the Frst World War. 1 Determne the angle (between and 9 degrees) that gves the maxmum range or "Parskanone". For ths angle calculate tme o lght and max alttude o the shells. For the angle calculated n the rst part, study the eect o ar densty, and varable ar densty on the trajectory (thus you run three calculatons: no ar resstance, ar resstance wth constant ar densty, varable ar densty). 3 Dscuss the accuracy o "Parskanone,.e. how much the ollowng eects would aect the accuracy: varatons n the ar densty (day/nght temperature, ran), wnd, and ntal speed. 4 Extra credt. Utlze an adaptve step-sze control. Use ether the doublng technque wth 4 th order Runge-Kutta, or Fehlberg s 5 th order Runge-Kutta wth error estmaton. Reerence normaton: The shell mass - 94 kg., ntal speed - 16m/s, calber - 1 mm, and the C coecent s about.1. Approxmate the densty o the atmosphere as ρ = ρ *exp(-y/y ), where y s the current alttude, y = 1.*1 4 m, and ρ s ar densty at sea level (y=). 3
Project 3: Random walks n two dmensons (Due on Monday, Aprl 5, 1 by 13:3) Part 1: Duson (smple random walk). Wrte a program that smulates a random D walk wth the same step sze. Four drectons are possble (N, E, S, W). Your program wll nvolve two ntegers, K s the number o random walks to be taken and N s the maxmum number o steps n a sngle walk. Run your program wth at least K >= 1. Fnd the average dstance R to be rom the orgn pont ater N steps. Plot the mean dstance travelled R versus the number o taken steps. Assume that R has the asymptotc dependence as R~N α, and estmate the exponent α. Extra credt (1 pont): consder a smple random walk n 1D (two drectons o moton) and 3D (three dmensons, sx drectons o moton). What wll be the exponent α or 1D and 3D. Compare your results wth the D case. Part : Random walk on a D crystal. Consder a two dmenson lattce o sze L*L. Randomly place a "random walker" on the lattce and start walkng (only our drectons are possble: let, rght, up, down). As soon as the random walker reaches a ste outsde the L*L area the random walk stops. Fnd the average number o steps S to get out o the crystal. Is there a connecton between S and L? Part 3: Random walk on a D lattce wth traps. Consder the same two dmenson lattce o sze L*L. Now the lattce contans a trap. (An analogy would be a cty wth (L-1)*(L-1) blocks and a polce patrol). Randomly place a "random walker" on the lattce. When the walker arrves to the trap ste, t can no longer move. So, the random walk stops when the walker ether trapped or out o the L*L area. Fnd probabltes to get the walker trapped and to go ree. Fnd also the mean number o steps ( survval tme ) beore a trap ste s reached, or the walker s out o the area, as a uncton o L. Explore ollowng scenaros: 1 A statonary trap located at the center o the area,.e. wth the coordnates (L/,L/) A randomly placed statonary trap 3 A randomly movng trap the trap walks randomly wth the same speed (one block n a tme). Snce the trap can not leave the L*L area, when needed, use ether the restrctve random walk or the patrol, or the perodc boundary condtons. 4 Extra credt ( ponts). There are two randomly movng traps. How wll t aect the probablty to capture the random walker. 5 Extra credt ( ponts). A persstent sngle trap the trap moves along a closed path (a box around the center wth a sde S<L lke a movng polce patrol). Does the outcome depend on the S/L rato? 4
Project 4: Sheldng a nuclear reactor (Due on Wednesday, Aprl 14, 1 by 13:3) Durng the World War II scentsts n Los Alamos (Manhattan project) had to nd how ar neutrons would travel n derent materals. Results were mportant or the calculaton o crtcal masses as well as sheldng. The physcsts knew most o the basc data and ther dependences on the neutron energy, namely, the average dstances between collsons o a neutron wth an atomc nucleus, the probabltes o neutron elastc or nelastc scatterng, probablty o capture by an atomc nucleus, the energy loss o the neutrons ater each collson. However, t was not clear how to use all ths normaton to nd a soluton. Ulam and von Neumann solved the problem by a novel numercal approach.e. smulatng a path o a neutron usng random numbers. Ths project below s a very smpled verson o the orgnal problem solved by Ulam and von Neumann. A beam o neutrons bombards a reactor's wall. Consderng moton o neutrons as a random walk on (x,y) plane nd probabltes or neutrons (as a uncton o the sheld sze) a) to be back n the reactor, b) to be captured n the sheld, c) to get through the sheld. Condtons: 1 only our drectons o moton are possble (let, rght, up or down) on the next step the neutron can not step back, but only orward, let or rght, and 3 the probablty to go orward s two tmes more than changng a drecton 4 on each step the neutron looses one unt o energy 5 ntal neutron energy s enough or 1 steps 6 ntal neutron velocty s perpendcular to the sheld 7 a capture probablty on every step s.1 Assume that the probablty P to get through the sheld has the asymptotc exponental dependence as P~e -ax wth x s the sze o the sheld. Estmate the exponent a. Optonal (bonus ponts): Consder the ntal neutron energy as a normal dstrbuton wth a mean value o 1 steps, and a standard devaton o steps Note: the sheld's sze s measured n "steps", where one step corresponds to an average dstance that neutrons move between collsons. 5