Example 3.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.  Assume that air resistance is proportional to the square of 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 3.

Notice that the maximum altitude will occur when the time is near t = 7,
and the arrow will hit the ground when the time is near t = 14.

Notice that the ascent time is the same as the descent time.
We desire to have a model in which the descent time is larger that the ascent time.
We expect this to happen in the "real world."

Remark. No matter how much you like the above model, it isn't right.  Indeed, the velocity function  V[t]  has a vertical asymptote at  

Graph the V[t] and the vertical asymptote.

Find the limiting velocity for  V[t].

Do we really believe that the limiting velocity for  V[t]  is   ?

Model for the Ascent

We can use the above solution for the ascent portion of the curve only.

Remark. With air resistance the descent time must be greater than the ascent time!  The D. E. for the descent must have the sign of the term with    positive. Let's call the velocity function  U[t]  on the descent portion of the curve.

Model for the Descent

We muse use the differential equation    for the descent portion of the curve.

Combine the Ascent and Descent

We expect that the limiting velocity for U[t] should be finite !  

Notice that the maximum altitude will occur when the time is near t = 7,
and the arrow will hit the ground when the time is near t = 15.

In this model, the descent time is larger that the ascent time.
It took quite a bit of effort to make an ascent function and a descent function !

Compare the three models of examples 1, 2, and 3.

The Runge-Kutta solution.

Now 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 how easy it is to obtain a Runge-Kutta solution.  

Often times there is not analytic solution and a numerical solution must be used.

(c) John H. Mathews 2004