Old Dominion University Physics 811 Fall 2008

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1 Old Dominion University Physics 811 Fall 008 Projects Structure of Project Reports: 1 Introduction. Briefly summarize the nature of the physical system. Theory. Describe equations selected for the project. Discuss relevance and limitations of the equations. 3 Method. Describe the algorithm and how it is implemented in the program. 4 Verification of a program. Confirm that your program is not incorrect by considering special cases and by giving at least one comparison to a hand calculation or known result. 5 Results. Show the results in graphical or tabular form. Additional runs can be included in an appendix. Discuss results. 6 Analysis. Summarize your results and explain them in simple physical terms whenever possible. 7 Critique. Summarize the important concepts for which you gained a better understanding and discuss the numerical or computer techniques you learned. Make specific comments on the assignment and your suggestions for improvements or alternatives. 8 Appendix. Give a typical listing of your program. The program should include your name and date, and be self-explanatory (comments, structure) as possible. Project 1: Simple projectile motion (Warming up project) (Due on September 16, 008 by 13:30) Consider a simple case of two-dimensional projectile motion: no air resistance. Equations of the motion are x y f f = x + ( v i = y i 0 + ( v 0 cosθ ) t sinθ ) t gt / 1 Calculate and plot the trajectory of the projectile motion for a given initial conditions (angle, speed and position). You may accept that y f =0.0 Find numerically the angle corresponding to the maximum horizontal range of the projectile for initial y i = 0.0. Calculate for this angle the maximum range, and time of flight. Present results in tabular and graphical forms. 3 Find how the angle corresponding to the maximum horizontal range of the projectile depends on the initial height. Select the initial speed of the projectile as of 100.0m/s, and the free fall acceleration as of 9.81m/s. 1

2 Old Dominion University Physics 811 Fall 008 Project. Data fitting (Due on October 7, 008 by 13:30) Cross sections of ionization by electron and ion impacts have been a subject of extensive studies as well as applications. Atomic data are collected in various international and national atomic data bases. A list of most popular atomic data bases can be found on this web page However, raw data are not practical for use in computer simulations. Therefore for practical applications it is common to fit available data to known formulae or to use some semi-empirical equations. More information on the subject can be found in multiple papers and reviews. The goal of this project is to fit experimental cross sections of electron impact ionization of the Neon ion (Ne + ) to a couple known formulae. Here are the equations: 1 σ ( E) ln IE A E = + I N k = 1 B k 1 I E k (1) where E is the energy of the incident electron, I is the ionization potential, A and B k are the fitting coefficients, E σ( E) = A ln B tan IE + I 1 1 E I β I () where the fitting parameters are A, B and β. Details and explanations for the equations above can be found in papers [1-]. Fitting experimental data from a table below consider various values of k (k = 1,, 3). You may be interested to apply your knowledge of error analysis to the problem. 1 K. L. Bell, H. B. Gilbody, J. G. Hughes, et al J. Phys. Chem. 1, 891 (1983). A. L. Godunov and P. B. Ivanov Physica Scripta. Vol. 59, (1999)

3 Old Dominion University Physics 811 Fall 008 Table.1 Some of published results for the single ionization cross sections of Ne + by electrons, where the impact energy is in ev and cross sections are in cm. Dolder et al. (1963) Donets & Ovsyannikov (1981) Dieserens et al. (1984) E σ E σ E σ

4 Old Dominion University Physics 811 Fall 008 Project 3. Projectile motion with air resistance (Due on October 8, 008 by 13:30) Write a program that simulates the projectile motion with allowing for air resistance, varying air density and wind. The air resistance force on an object moving with speed v can be approximated by F drag =-0.5Cρ 0 Av, where ρ 0 stands for air density (ρ 0 =1.5 kg/m 3 at sea level), and A is the cross section. The drag coefficient C depends on an object shape and for many objects it can be approximated by a value within Use Runge-Kutta method as a primary method for solving a system of differential equations. Extra credit: compare accuracy of Verlet method to Runge-Kutta. Part 1. Study the trajectory of shells of one of the largest cannons "Pariskanone" used during the First World War. Calculate effects of air resistance, varying air density and wind on the range, time of flight and max altitude of shells. Determine the angle (between 0 and 90 degrees) that gives the maximum range for such cannon. Discuss the accuracy of the "Pariskanone". Some initial information: The shell mass - 94 kg., initial speed m/s, caliber - 10 mm, and the C coefficient is about 0.1. Approximate the density of the atmosphere as ρ = ρ 0 *exp(-y/y 0 ), where y is the current altitude, y 0 = 1.0*10 4 m, and ρ 0 is air density at sea level (y=0). Part. In standard university physics textbooks it is assumed that the air, through which projectiles move, has no effects on projectile motion. Let's consider whether we need to revise some well known to us statements if air resistance is taken into account. Statement 1: "Time to move up is equal to time to move down". Suppose an object is thrown vertically upward with an initial velocity. How does the time of ascent compare to the time of descent if air resistance is taken into account? Statement : "Two bullets, one fired horizontally and one dropped vertically from rest at the same instant, reach the ground at the same time". Compare the motion of two identical objects that both start from the same height h. One object is dropped vertically from rest and the other is thrown with a horizontal velocity v. Which object reaches the ground first if air resistance is included? Before doing your computer simulations give reasons for your anticipated answers in each case. Compare later your anticipated answers with results of your computer simulations. For the simulation select any object that you like (but no cats, dogs, etc.). The drag coefficient C for a specific object can be found either in the lecture notes, or in publications, or on the Web. The initial velocity v should be fast enough to observe effects of air resistance, but realistic. 4

5 Old Dominion University Physics 811 Fall 008 Project 4. Random walks (Due on November 6, 008 by 13:30) Part 1. Study quality of a built-in random number generator (in C++ or Fortran): evaluate 3-rd moment of the random number distribution, evaluate near-neighbor correlation, plot a D distribution for two random sequences x i and y i (a parking lot test), plot a D distribution for correlation (x i, x i +4). How the 3-rd moment and near-neighbor correlation do depends on the number of random numbers in your study? You may start from 4 random numbers and increase by on each step. Part. Write a program that simulates a random D walk with the same step size. Four directions are possible (N, E, S, W). Your program will involve two integers, K = the number of random walks to be taken and N = the maximum number of steps in a single walk. Run your program with at least K >= Find the average distance R to be from the origin point after N steps. Assume that R has the asymptotic N dependence R~N a and estimate the exponent a. Part 3. Randomly place the "random walker" on a two-dimension lattice of L sites in a row (like a city with L+1 blocks in a row). Consider the "random walker in a city" as a persistent random walk (on each step the walker cannot step back, only forward, left or right). Find the average distance D (or the average number of steps) in order to be out of the city limit. Is there a connection between D and L? Part 4. Suppose a police foot patrol wants to arrest the random walker not knowing his position ahead of time. What tactics would increase their chances to capture the random walker a) staying at one place waiting for the walker to come, b) random walk from a random point, c) following the same closed path (a square one) inside the city d) following a closed path along the city limits? The speed of the foot patrol is equal to the speed of the walker. Extra credit: Can the random walker affect his chances to be arrested by visiting a bar from time to time and staying there for k turns? The patrol checks bars on each intersection. Would it be different if the patrol does not check bars? 5

6 Old Dominion University Physics 811 Fall 008 Project 5. Classical collisions (Due on November 0, 008 by 13:30) In classical scattering theory the differential scattering cross section is related to the impact parameter and the scattering angle as dσ b = dω sin( θ ) db dθ Part 1: Rutherford scattering. Consider the elastic scattering of a projectile with mass M 1 and electric charge Z 1 with a target of mass M and charge Z. The target is fixed in space. The projectile-target interaction is describe by the Coulomb potential that can be written in atomic units as Z1Z ( R) = R V The analytic solution for the differential cross section is the Rutherford scattering formula. dσ Z1Z = dω 4E sin ( θ / ) Solve the problem of the Rutherford scattering numerically as an ODE initial value problem in the (x,y) plane. You initial conditions are the projectile velocity and the impact parameter. Calculate the differential cross section and compare your numerical result to the Rutherford scattering formula. Part : Rutherford scattering (modified). The target is initially at rest, but may move during the collision due to the projectile-target interaction. Study the effect of the target motion on the differential scattering cross section. Now you have two moving objects. You should test your solutions by calculating the total energy and angular momentum of the system. At any moment of the collision the total energy and the total momentum must be conserved. Sure that the effect would depend on the ratio M 1 /M. Part 3: Yukawa potential (screened Coulomb potential). The long range Coulomb potential has been a serious problem in quantum scattering theory. Study numerically the elastic scattering of two particles interacting by the Yukawa potential Z1Z ( R) = exp( αr) R U Test you results using conservation of the total energy and the total momentum for the two particles. Compare you numerical results with the analytic solution θ = π r min ( b / r 1 ( b / r) ) U( r) / E dr But first check conditions when the equation above is valid. Compare your results for the two potentials (Coulomb and Yukawa). Part 4: Optional. Calculate the total cross section of the elastic scattering for the parts 1-3. Comments: 1) It is much more convenient to work in the atomic system of unit. ) You need to use following numerical procedures: numerical solution of the ODE initial value problem, numerical derivatives, and numerical integration 3) Be reasonable selecting initial conditions, as well as M 1, Z 1, M, Z, and α. 6

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