Projects assignments (Fall 2006)

Project Reports

Introduction. Briefly summarize the nature of the physical system.
Theory. Basic equations (physics)
Method. Describe the algorithm and how it is implemented in the program.
Verification of 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.
Results. Show the results in graphical or tabular form. Additional runs can be included in an appendix.
Analysis. Summarize your results and explain them in simple physical terms whenever possible.
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.

Project 1: Projectile motion

Due on October 16th, 2006 by 16:00

Figure shows a navy ship 1 mile (1609 m) from a fort defending the harbor entrance of an island. The fort is located 40 meters above the sea level. The armory house is about 70 meters beyond the fort walls, which are 35 meters high. The ship fire cannons at the muzzle speed v= 180 m/s.

  1. At what angle from the horizontal must the cannons be fired to hit the armory.
  2. What is the minimal muzzle speed to hit the armory? At what angle it may happen?
  3. Is it important to take into account that the ship is actually moving toward the fort with the speed about 5 knots?
    (1 knot = 1 nautical mile/hour = 1.852 km/h).

Solutions: a fortran code proj01.f and a file with solutions proj01.dat

Project 2: Numerical model for projectile motion with air resistance

Due on October 25th, 2006 by 16:00

Write a program which uses 4th order Runge-Kutta method to solve the problem of a projectile motion with air resistance to determine position as well as corresponding velocity as a function of time in (x,y) plane. Initial data for the program are initial velocity, the inclination angle, the projectile mass, and appropriate constants to calculate the drag force.

You have to carefully outline the problem before writing a computer code.

Application: Using the code study the effect of air resistance on the path of a cannon ball.
a) Calculate the range and maximum height at sea level for a cannon ball with initial speed of 180.0m/s at different initial angles (15, 45 and 75 degrees).
b) Determine the angle (between 0 and 90 degrees) that gives the maximum range. Does your answer depends on the effect of air resistance.

Test your program for the projectile trajectories ignoring air resistance (good test, because you may solve the problem analytically)

Assume:
The drag force for a cannon ball can be approximated by Fdrag = -CpAv2 with C=0.1, p stands for air density (po=1.25 kg/m3 at sea level), and A is the cross section. The density of the atmosphere varies as p = po*exp(-y/y0), where y is the altitude, y0 = 1.0*104m, and po is air density at sea level.(y=0).
The diameter of the cannon ball is about 20.0 cm.
The weight is about 10.0 kg

Bonus question:
Consider the range and time of flight one of the largest cannons "Pariskanone" used during the first World War. Study the effect of the varying air density.
The shell mass - 94 kg.,
Initial speed - 1600 m/s
Caliber - 210 mm
C coefficient is about 0.06

Solutions: Calculations for 180 m/s (proj02a.pdf), max range (xrange.dat), WWI (proj02b.pdf)

Project 3: Random walks in two and one dimensions

Due on November 8th, 2006 by 16:00

Write a program that simulate a random 2D walk with the same step size . Four directions are possible (N, E, S, W).
Your program will involve two large 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 >= 1000

  1. Find the average distance R to be from the origin point after N steps
    Assume that R has the asymptotic N dependence R~Na and estimate the exponent a.
  2. 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). Find the average distance D (or the average number of steps) to go to be out of the city limit. Is there a connection between D and L?
  3. Synchronized random walks. Randomly place two walkers on a one-dimension lattice of m sites, so that both walkers are not in the same site. Two random walks can move on a one-dimensional line with m sites. In each step a direction, either left or right, is chosen randomly. Then both random walkers move in this direction. If one random walk hits the boundary, it is reflected, i.e. the random walk remains at its site on the left or on the right. As the other random walker is not affected, the distance d between both decreases by 1 at each reflection. Otherwise d remains constant. A trial end when both walkers are at the same site. Find mean time for two walkers to reach the same site. What is connection between the mean time and the number of sites m? (The model is relevant to a method of doing cryptography using neural networks)

Solutions: 1) R~N0.5 2) D~L2 3) The average number of steps until synchronization increases with m2, the square of the system size. (see Ruttor et al J. Phys. A: Math. Gen. 37 (2004) 8609-8618)

Calculations for 3 projects: proj03.pdf

Project 4: Polymer physics and other random processes

Due on November 20th, 2006 by 15:55

A polymer consists of N repeat units (monomers) with N>>1. For example polypropylene can be represented as ... -CH2-CH2-CH2-... Random walk models can be used to study global properties of polymer formation. Polymers have an important physical feature, that is, two monomers cannot occupy the same spatial condition (the excluded volume condition). For random walk models it corresponds to the self-avoiding walk (SAW). This model consists of the set of all N-step walks starting from the origin subject to the global constrain that no lattice site can be visited more than once in each walk: this constrains accounts for the excluded volume condition.

Consider a 2D square lattice and write a program and compute
a) average number of monomers in this kind of 2D polymer
b) the fraction of successful attempts f(N) at constructing polymer chains with N total monomers (plot a histogram)
c) what is the maximum value of N that you can reasonably consider?
d) plot an example for a polymer with an average number of monomers in your simulation.

Solutions: a) about 72 monomers, b) histogram, c) about 240 monomers, d) example 1, example 2, example 3

Part 2. (Bonus)

The gambler's ruin problem. Suppose that a person decides to try to increase the amount of money in his/her pocket by participating in some gambling. Initially, the gambler begin with $m in capital. The gambler decides that he/she will gamble until a certain goal, $n (n>m), is achieved or there is no money left (credit is not allowed). On each throw of a coin (roll of the dice, etc.) the gambler either win $1 or lose $1. If the gambler achieves the goal he/she will stop playing. If the gambler ends up with no money he/she is ruined.

Write a program to compute
a) What are chances for the gambler to achieve the goal as a function of k, where k=n/m?
b) How long on average will it take to play to achieve the goal or to be ruined?

Solutions: see the graph

Project 5: Motion of planets

Due on November 29th, 2006 by 15:55

Write a program to simulate the Earth's orbit around the Sum
a) Considering the special case of a circular orbit v0=(GM/r)1/2.
b) Calculate motion with initial velocities: v = 0.8, 1.1, 1.2, 1.3, 1.4 v0.
c) Check conservation of energy* with time.
d) Check conservation of angular momentum*.
e) Find the numerical value of the period for each v.
f) Find the numerical value of eccentricity for each v

* It is sufficiently to calculate the energy and angular momentum per unit mass, i.e. E/m and Lz/m

Solutions: data and a graph

Final Project

Due on December 6th, 2006 by 16:00 (presentation in class and report)

Every advantage in the past is judged in the light of the final issue.
Demosthenes (384 BC - 322 BC)

Chose one of problems for your final project.

Problem 1: A mini-solar system with two planets

Write a program to simulate the motion of two planets (for example Earth and Venus) around the Sun.
Study the effect of interaction between planets on their orbits.
Are there total energy and angular momentum of two planets conserved?
Are the energy and angular momentum of planet 1 conserved?
What if the force of the Earth-Venus interaction was 1000 times stronger?

* It is sufficiently to calculate the energy and angular momentum per unit mass, i.e. E/m and Lz/m

Interesting link: Millennium Simulation - the largest N-body simulation carried out thus far (more than 1010 particles).
A 3-dimensional visualization of the Millennium Simulation shows a journey through the simulated universe
http://www.mpa-garching.mpg.de/galform/millennium/

Problem 2: The classical helium atom

Write a program to simulate motion of electrons in the 2D classical helium atom. Study the effect of interaction between electrons on their orbits.
Are there total energy and angular momentum of two electrons conserved?
Are the energy and angular momentum of electron 1 conserved?
What if the force of electron-electron interaction was 100 times weaker?

Helium Atom as a Classical Three-Body Problem,
T. Yamamoto and K. Kaneko Phys. Rev. Lett., vol. 70, 1928 (1993)

Problem 3: Double stars

Consider a planet orbiting about two fixed stars of equal masses. Study types of orbits depending on initial conditions (velocity and position).

Problem 4: Effect of a solar wind

Assume that a satellite is affected not only by the Earth's gravitational force, but also by a weak uniform "solar wind" of magnitude W acting in a horizontal direction. Choose initial conditions so that a circular orbit would be obtained for W=0. Then chose a value of W/m whose magnitude is about 3% of the acceleration due to gravitational field and compute the orbit. Consider W=0 when the Earth blocks the solar wind, i.e. is the satellite is in the Earth’s shadow.
What is the effect of the solar wind on the satellite orbit?
How does the total energy and total angular momentum change?

Problem 5: Physics and sport

The goal of the project is to study effects of the air drag force, aerodynamic drag crisis, and Magnus force on the flight of various balls used in sport.

Assume:

Type of ball
speed (m/s)
diameter (cm)
mass (kg)
Reynolds number
baseball
45.0
7.32
0.145
2.08e+5
basketball
9.0
24.26
0.6
1.46e+5
golf
60.0
4.26
0.046
1.73e+5
soccer
30.0
22.2
0.454
4.31e+5
tennis
50.0
6.5
0.058
1.96e+5

The drag force: Fd = -0.5CdpAv2, where p stands for air density (p=1.25 kg/m3 at see level), A is the cross section, v is the speed. Generally, the drag coefficient Cd dependes on the Reynolds number (see figure from Frohlich Am J. Phys. 52, 325 (1984)).

The Magnus force (spin effect) can be approximated as: Fm = -0.5CmpAv2, where the lift coefficient Cm depends on angular speed. The coefficient Cm for various balls can be found in following papers:

Baseball: A. Rex Am. J. Phys. vol. 53, 1073 (1985), R. Adair The Physics of baseball. New York, Harper and Row (1990)

Tennis: A. Stepanek Am. J. Phys. vol. 56, 138 (1988), J. Zayas Am. J. Phys. vol. 54, 622 (1986), C. Lindsey et al The Physics and Technology of Tennis, USRSA (2004)

Golf: H. Erlichson Am. J. Phys. vol. 51, 357 (1983), T. P. Jorgensen The Physics of Golf, American Institute of Physics (1999)

Basketball: P. Brankazio Am. J. Phys. vol. 49, 356 (1981)

Soccer: J. Wesson The Science of Soccer, Institute of Physics Publishing (2002)