P2820 Final W2011

April 15, 2011   9-11am	30 marks

There are 3 questions, each worth 10 marks.
Note: Do not worry about axes labels for plots unless asked specifically to do so, but do and include units for physical quantities otherwise (give numerical answers to three significant figures).

Question 1 [10 marks]

a) Download the file vt.dat from the course website.  It contains two colums of data: column 1 gives time in seconds and column 2 gives velocity in m/s.  Import the data and plot the velocity as a function of time.

b) Use the forward difference scheme directly on the data to construct a new table of data containing the acceleration as a function of time.  Plot the acceleration as a function of time.  (Note: the data here are not evenly spaced in time, but the forward difference method presented in the course will still work.)

c) From the table of acceleration that you constructed, find the average acceleration.

d) Find the line of best fit through the velocity data.  Plot the line with a thick red line overtop of the data in a single plot.  Include axes labels.

e) What is the value of the acceleration obtained from the linear fit?  What is the percentage difference between this value and the average value of the acceleration obtained in part (c)?

Question 2 [10 marks]

The potential energy of a particle as a function of distance x (m) from a particular wall is given in J by

V(x)=(1/(x^5))-(1/(x^3))

a) Plot the potential energy as a function of x.  Ensure that your graph looks reasonable in terms of showing the features of the potential (i.e. plot over a reasonable range of x, and use an appropriate scale).

b) Define a function F(x) that returns the force on the particle in terms of x.  (Remember, “force is minus the gradient of the potential energy”.)  Plot F(x) in a reasonable way.

c) At what value of x is the force zero?  At what value is the force the most attractive?

d) Solve for the trajectory of a particle of mass 1.5kg initally at rest and at a distance of 1.1m from the wall.  Find and plot the trajectory for the first 50s.

e) Plot the potential energy of the particle as a function of time.  Find the average value of the potential energy over the first 50s.

Question 3  [10 marks]

A 1.8 kg projectile is fired straight up from the ground with initial speed of 50m/s.  The magnitude of the drag force is k (v^2), i.e. proportional to the square of the speed, and k=0.3 kg/m.  Acceleration due to gravity is g=9.8 m/s2.

Write your own numerical ODE solver employing 4th order Runge-Kutta to find the height of the projectile as a function of time for the first 4 seconds.  Use a time step of 0.1s.  

Plot the height and velocity obtained from your method as funtions of time.  

At what time does the projectile return to the ground?