P2820 FINAL   APRIL 18, 2009   3-5pm

This is an open book final.   There are 4 question, for a total of 40 marks.  Time allowed is 2 hours.

Name:
Student #:

Q1 (8 marks)

Baffo Boy, not quite a superhero yet, runs off a tall cliff on his self-made hanglider with initial horizontal velocity 2 m/s (zero vertical velocity).  In the vertical direction, the hanglider provides a net upwards force in newtons of 445 (Subscript[v, x]), where (Subscript[v, x]) is the horizontal component of the velocity in m/s.  There is also a drag force of -0.2 v, where v is the velocity (resulting in forces in both vertical and horizontal directions).  The combination of glider and Baffo Boy have a mass of 90kg.  Use g=9.8 m/(s^2).

a)

Find Baffo Boy's trajectory using a built-in Mathematica ODE solver to for the first 14 seconds and plot your result.

b)

What maximum height relative to his starting point does Baffo Boy achieve?

Q2 (12 marks)

For this question you will need to download the file called trajectory.dat from the course website.  The first column gives time in seconds, while the second column gives the position of a falling particle in metres.

a)

Import the data and plot them nicely.

b)

Use the centred difference scheme for the first derivative to find the velocity at all times where the method allows you.  Plot the velocity you obtain.

c)

Use the centred difference scheme for the second derivative to find the acceleration directly from the positions at all times that the method allows.  Plot the acceleration you obtain.

d)

Assuming that the magnitude of the drag force is F = k (v^2) (where v is the velocity) determine statistically the drag constant k. The mass of the object is m=4.5kg and g=9.8 m/(s^2). (Hints: Newton's second law; linear fit).

Q3 (12 marks)

A frictionless oscillator consists of a mass of 1.7 kg attached to a massless nonlinear spring.  The magnitude of the restoring force of the spring is given by  k (x^3) with spring constant k=3 N/(m^2), where x is the displacement of the spring from equilibrium.

a)

Use a built-in Mathematica ODE solver to find and plot the position of the mass as a function of time from t=0 to t=10s.  The initial displacement is 0.2m and the initial velocity is 11m/s.

b)

Find the period of oscillation of this oscillator from the roots of x(t).

c)

What is the average value of (x^2) during this time interval?

d)

Write your own code that uses the 4th order Runge-Kutta algorithm to find the position as a function of time for t=0 to t=10s.  Use a step size of 0.1s.  Plot your result along with your result from part a) on the same graph.

Q4 (8 marks)

Model a particular vibrating string by solving the wave equation on x ϵ [0,2]metres and t ϵ [0,5] seconds, where the phase velocity is 0.7 m/s, with the following initial and boundary conditions.
- displacement is initially zero
- velocity is initially zero
- left endpoints is fixed at zero displacement
- right endpoint's displacement is given by f(t)=exp[-(((t-4))^2)] sin(2t)
Plot the displacement of the string at t=10s.