Project 6: Skydiver with no parachute, no air resistance

Wednesday, April 16, at 4:00 PM: Pseudocode
Monday, April 21, at 9:00 AM: The finished MATLAB code and analysis

Your job in this project: use Euler's method to determine the height and speed of a man who jumps out of an airplane.

The situation is: an airplane moves in level flight at a height of H = 10,000 meters above the ground. Joe opens the door and steps out at t = 0. What are his

height y(t)
velocity v(t)

as a function of time?

You should use Euler's method in this project. Apply a first-order expansion of both y(t) and v(t) around the current time t0, to predict the height and velocity at the next time t1.

There is an equation which gives the velocity at time t1, in terms of quantities known at time t0.

        v(t1)  =  v(t0)  +   v'  * (t1 - t0)

               =  v(t0)  +  (-g) * (t1 - t0)

since the first derivative of velocity is acceleration, which is simply -g = -9.8 m/s^2.

Write down the similar equation which gives the height y(t1) as a function of quantities known at time t0; use only first derivatives, as required by Euler's method.
You need a set of initial conditions for the problem: the values of height, velocity, acceleration, and time at the start. What are they?

Calculating position as a function of time

Write a MATLAB routine which calculates position and velocity as a function of time for this problem. It should look like this:

  function skydiver(start_height, timestep, output_file)

where

      start_height            is the starting height, in meters
      
      timestep                is the size of the timestep (t1 - t0) to use 
                                  in calculations, in seconds

      output_file             is the name of a file in which you will
                                  write values of time, height, and speed

As you go through a loop, calculating height and speed at each time, you should write the values into a text file, for later use. Here's how to do it:

       % (near the start of the program)
       fid = fopen(filename, 'w');


       % (in the main loop)
           fprintf(fid, '    0.0000     0.0000     0.0000 \n', time, height, velocity);
       % (end of loop)


       % (near the end of the program)
       fclose(fid);

Examples of simple functions which read and write data from and to files can be found on the course WWW page:

The output file should contain data like this:

  0.0          10000.0         -0.0
  1.0           9995.1         -9.8
  2.0           9980.4        -19.6

where the first column is time (in seconds), the second column is height above the ground (in meters), and the third column is velocity (in meters per second, upwards positive and downwards negative).

Use a timestep of 1.0 seconds. Stop the loop when Joe reaches the ground.

At what time does Joe reach the ground, according to your MATLAB routine? How fast is he going?
Solve the problem analytically. At what time does Joe hit the ground, and what is his final speed?
Compare the analytic answer to the MATLAB answer. Are they exactly the same? If not, explain.
Re-run the MATLAB routine using timesteps of 0.1 seconds and 0.01 seconds. Again compare MATLAB to theory, and discuss.

Making a plot of position or velocity versus time

Write a routine which reads a datafile produced by your MATLAB routine (containing time, height, and velocity), and creates plot(s) showing height and velocity as functions of time. It should look like this:

    function  plot_skydiver(datafile, y_var)

where

            datafile           is the name of a file in which the data
	                           for the simulation have been placed

            y_var              is a string which indicates the value
	                           to plot versus time: it must be either
				         'height'
                                   or
				         'velocity'

To get you started, look at a program I wrote which does most (but not all) of the tasks:

plot_from_file.m

You can copy this program and then modify it as necessary.

You'll probably want to know how to compare two strings in MATLAB; that is, how to check if the user supplied 'height' or 'velocity' as the second argument to your function. You can use the strcmp function to do the trick, as illustrated in this sample function:

guess_letter.m

Your program should produce a graph which looks something like this:

Bells and Whistles

There is a small decrease in the gravitational acceleration of a person who is far above the surface of the Earth. Assume that at the location of Joe's jump, the gravitational acceleration at the surface is exactly 9.80 m/s^2.
```
                          G*M
               accel  =  ------  =  9.80  m/s^2
                          R*R
     
```
Now, modify the formula so that it yields the acceleration of an object at some height H above the surface. Expand the formula so that it shows the acceleration to first order in H/R, assuming that H << R. Use this formula to calculate Joe's acceleration as he falls; it will not be constant, but grow very slightly as he comes closer to the ground. Use a value R = 6380 km in your calculations.
How big a difference does this variable acceleration make in the time it takes Joe to hit the ground?
Make a slightly modified version of the plot_skydiver function in addition to the ordinary one. In this modified version (call it movie_skydiver), your graph should start off empty, and then gradually show Joe's position, one dot after another. In other words, make a movie showing Joe's height as a function of time. Try to make the dot appear in real time; that is, if it takes Joe 30 seconds to hit the ground, plot the points so that 30 seconds after he jumps, the last dot appears at the bottom of the graph.
You can do this in several ways. One is to figure out how to pause the program's execution for just the right amount of time between dots. Another is to use MATLAB's movie functions to create a movie which plays one frame per second, and draw the appropriate dot for each second. You can find some very terse information on making MATLAB movies at
http://www.nwl.ac.uk/staffpages/cjsc/MATLAB_Movies/movies.html

This page maintained by Michael Richmond. Last modified Apr 13, 2003.