PHYSICS 542 / Applications of Numerical Methods in Physics

R. J. Wilkes

Exercises #3

Not to be handed in; answers (with set #4) will be posted next TUESDAY.

1. In an experiment, data from a gas flowmeter are logged as follows. Find the total gas consumed in the 10 minute interval, with error < 0.001 liter.

time, min                                   flow rate, liters/min

0                                              0.260

1.25                                         0.208

2.5                                           0.172

3.75                                         0.145

5.0                                           0.126

6.25                                         0.113

7.5                                           0.104

8.75                                         0.097

10.0                                         0.092

2. Use the trapezoid and Simpson's Rule methods to integrate sinc(x)=sin(x)/x fromx=0 to x=0.8, using panel width h=0.2.

3. Try using Gaussian Quadrature with N=4,8,16 on the Runge function, R=1/(1+25x2), from x=-1 to x=1.

4. Three boxes contain respectively 1) two gold coins, 2) one gold coin + one penny, and 3) two pennies.  Each box has its 2 coins in separate closed compartments. If you select a box at random, and the first compartment contains a gold coin, what is the probability that the second compartment will also contain a gold coin?

5. A supplier says pieces of aluminum beam will be delivered with mean length 80 cm and standard deviation 8 cm. What probability distribution is likely to apply? Assuming that distribution, what percentage of the pieces will be longer than 68 cm? What length limits apply to 99% of the pieces?

6. Pions are subatomic particles that promptly decay to neutrinos and muons. Pions are abundantly produced when a high energy proton beam hits a tungsten or carbon target.  To make a neutrino beam, we first make a pion beam, and then let it fly through an empty chamber (decay tube) of adequate length. A thick wall of earth following the decay tube stops the undecayed pions and most of the muons, but not the neutrinos, so only neutrinos (and a few muons) finally emerge. Suppose a pion beam is introduced into a decay tube of length 80 meters; for these pions, the mean distance traveled before decaying is 558 m. (Assume they all travel at speed pf light c, so survival distance is proportional to lifetime.) What fraction of the pions decay to neutrinos before running into the earth wall? Do you need to know how far upstream the pions themselves were produced, i.e. where the decay tube is located relative to the pion production target? (Hint: Poisson assumptions apply to radioactive decays.)

1. GJ, GE/BS and LUD require on the order of N3, N3/3, N3/6 operations respectively, so for 1 microsecond/op we get

N                                 time for: GJ                                           GE/BS             LUD

10                                1 msec                                                 0.3 msec          0.16 msec

100                              1 sec                                                    0.3 sec             160 msec

1000                            1000 sec (17 min)                                5 min                160 sec

10000                          1E6 sec (12 days)                                4 days              2 days

2. Condition number = 144, error on x <0.072.

4. f(x)= 1 - 1.4x + 0.4 x3                                                                      x=[0,1]

-0.2(x-1) + 1.2(x-1)2 - 0.5(x-1)3                                    [1,2]

0.5 + 0.7(x-2) - 0.3(x-2)2 + 0.1(x-2)3                                      [2,3]

f'(3) = 0.4