Example 2.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.  Assume that air resistance is proportional to the velocity, , and use the model  
        .  
Find the velocity and position as a function of time, and plot the position function.  Find the ascent time, the descent time, maximum height, and the impact velocity.

Solution 2.

First, compute the solution using the Runge-Kutta method for second order D.E.'s.

The solution we seek is the first coordinate in the 2D system.

Now we can plot the solution.

Compare the Runge-Kutta solution with the analytic solution.

Notice that the maximum altitude will occur when the time is near t = 8,
and the arrow will hit the ground when the time is near t = 17.

In this model, the descent time is larger that the ascent time.
Don't we expect this to happen in the "real world."

Compare the two models of examples 1, and 2.

(c) John H. Mathews 2004