Example 4.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.
Assume that air resistance is proportional to the power of the velocity,  ,   and use the model  
        .  
Find the velocity and position as a function of time, and plot the position function.

Solution 4.

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.

The graph should lie between the ones obtained in examples 2 and 3.
Let's see if this happens.

The graph does lie where it should lie.

We are done.  

Aside.  We can "try" to get an analytic solution.

Clearly, Mathematica is having a difficult time of it.
If the initial condition is not used then Mathematica can go a little further in the solution.

We should trust our Runge-Kutta solution.

(c) John H. Mathews 2004