Problem 29                Due on 4/25/01

        In this problem we integrate a
second order differential equation as
a system of first order equations.

        Consider the motion of a cannon ball
which is fired straight up into the air.
The initial velocity is given by Uo, the
mass is M, and the radius is R.  We want
to calculate how high the cannon ball gets,
and how long it takes before it hits the
ground again (e.g., how long do we have to
get out of the way).  The motion of the ball
is governed by Newton's equations of motion,
and in addition to the force of gravity we
also have an air drag.  This drag is given by:

F = - sign(U)*0.5*pi*R^2*U^2*dens_air

If the mass M is 3000g, the initial velocity
Uo is 2e4 cm/s, the density of air is 0.001 g/cm^3,
and the radius is 10cm, calculate the maximum
height and the time for the ball to hit the
ground again.

Use a high order algorithm (e.g., not Euler Method) to 
solve the problem, and determine the maximum height
and contact time accurately using an interpolation
algorithm on the appropriate points.  Add these
values to a plot of the trajectory.

Hint:  Just as in the example problem, this is integration
is most easily done if you define a function which takes
in the height and velocity as an array, and then returns
the derivatives as an array.  You can then integrate both
in vector form.