Assignment: Numerical Simulation
Problem 1
The Maclaurin series expansion for cos (x) is cos (x) = 1 - x2/2 + x4/4! - x6/6! + x8/8! - ...
Starting with the simplest version cos (x) = 1, add terms one at a time to estimate cos (π/3). After each new term is added, compute the true and approximate percent relative errors. Use your calculator or MATLAB to determine the true value. Add terms until the approximate error is below 0.5%.
Problem 2
A nice application of integration is to compute the distance z(t) of an object based on its velocity v(t). Take our bungee jumper as example the velocity is given as v(t) = (√gm/cd) tanh? {(√gcd/m) t}
Suppose that we had measurements of velocity at a series of discrete unequally spaced times during free fall. Use the above equation to synthetically generate such information for a 70-kg jumper with a drag coefficient of 0.275 kg/m. Incorporate some random error by rounding the velocities to the nearest integer. Then use cumtrapz to determine the distance fallen and compare the results to the analytical solution z (t) = (m/cd) ln? [cosh {(√g cd/m) t}]. In addition, develop a plot of the analytical and computed distances along with velocity on the same graph.