Reference no: EM132224598

Lamoureux Assignment -

Exercise 1 - Plot a few Bessel functions of the first kind, using your knowledge of Python. Plot for orders α = 0, 1, 2, 3 and choose a sensible range of values for the domain.

Exercise 2 - We had a model for the traffic jam model, with a jump in the velocity function. Replace the jump function for v(x) with a smooth, monotonic function that has a left limit of v_left and a right limit of v_right. Choose a function where it is easy to calculate the anti-derivative of slowness 1/v(x).

Do calculate the function S(x) as the anti-derivative of slowness 1/v(x), for your choice of v(x).

Hint: Rational functions won't work. You might want to consider functions like arctan, tan^-1(x) or hyperbolic tan, tanh(x).

Exercise 3 - Adjust the code in Exercise 2 to use your new definition of v(x) and the resulting S(x). Show that it works by making a few illustrative plots.

Exercise 4 - Kepler's third law of planetary motion says that the length of time it takes a planet to orbit the sun is proportional to its distance from the sun, raised to some (fractional) power. That is:

T = kR^α,

where T is the length of time for one complete orbit, R is the distance between the planet and the sun, α is a fixed power, and k is some univeral constant that works for all the planets around our sun.

Use Dimensional Analysis (Buckingham's Pi Theorem) to derive this result. Tell me what the value of α is.

Don't use calculus! (Although you may have seen this solved via differential equations in a calc or physics class.)

Hint: There is some ambiguity because of two masses involved (sun and planet). Newton knew that the mass of the planet does not matter to T, so you can assume this as well. Newton's universal gravitation constant G also enters into the problem -- you can look up what units it is measured in, on Google. Or you can figure it out yourself from the force formula for planetary attraction

Force = G(Mm/R²)

where M, m are the masses of the sun and planet.

You can also check your answer by looking up Kepler's laws on Google.

Exercise 5 - Make a table listing the 8 planets plus Pluto, their distance to the sun, and the period of their orbit.

Make a log-log plot of period versus distance, and check that you get a straight line with slope equal to α. (At least approximately.)

i.e. Taking logs of the equation T = kR^α gives

log T = log k + α log R,

which is the equation of a line in x = log R, y = log T.

Exercise 6 - Nuclear bombs, when exploded in the atmosphere, produce a large fireball that expands in the approximate shape of a sphere of some radius r = r(t), which is a function of elapsed time. The rate at which it expands depends on the energy E released by the bomb, the elapsed time t since the detonation, and the density ρ of the surrounding air.

Curiously, the actual air pressure is irrelevant, as it is so small compared to the explosive blast. Use dimensional analysis to find r(t) as a function of E, t, ρ. (times some fixed constant)

Exercise 7 - Can you animate the result in Exercise 6, showing some bomb blasts of various energies? Something pretty?