Module for the Flight Trajectory

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

Background

    This is a model for the flight path of an airplane that is flying in the wind.  For illustration purposes, assume that the wind velocity is w blowing due north and the speed of the plane is s.  Furthermore, suppose that the destination point is an airport located at the origin and the the starting point is  a  miles east of the airport at the point  (a,0).  The plane always maintains a heading directly toward the origin.  The system of differential equations for the components  {x[t],y[t]}  position of the plane are

          with     

Points {x[t],y[t]} on the curve will lie in the first quadrant where x[t]>0  and  y[t]>0.  

The three cases.

Case i.    If  c < w,  the plane's speed is less than the wind velocity, then   and it follows that that

.  In this case the plane will never arrive at the origin because .

Case ii.   If  c = w,  the plane will not arrive at the origin, instead it's limiting position will be  .

Case iii.  If  c > w,  the plane's speed is greater than the wind velocity, and the plane will arrive at the origin.  

Numerical solutions for Flight Trajectory

    Mathematica's procedure "NDSolve" can be used to solve the system of  D. E.'s

          with     

However, there are precautions that must be taken.  The time interval    over which the numerical computation is to be taken is not known in advance.  If too large an interval is given, then Mathematica's numerical subroutine will encounter a problem and the value must be reduced.

Example 1.  Find the flight path  {x[t],y[t]}  if the starting point is {1,0}  and  
the plane's speed is c=2 and the wind velocity is w=1 due north.

Example 2.  Find the flight path  {x[t],y[t]}  if the starting point is {1,0}  and  
the plane's speed is c=1 and the wind velocity is w=1 due north.

Example 3.  Find the flight path  {x[t],y[t]}  if the starting point is {1,0}  and  
the plane's speed is and the wind velocity is w=1 due north.

Analytic solutions for Flight Trajectory

Example 4.  Use symbolic techniques of differential equations to solve

          with     
Obtain "two" solutions.

        ,  
    and
        ,  

where  .

Example 5.  Find the flight path  {x[t],y[t]}  if  {x[t],y[t]] = {50,0}  and  
the plane's speed is and the wind velocity is w=50 due north.

Example 6.  Find the flight path  {x[t],y[t]}  if  {x[t],y[t]] = {1,0}  and  
the plane's speed is c=2 and the wind velocity is w=1 due north.

Example 7.  Use the flight path  {x[t],y[t]}  starting at the point is {1,0}  in example 1.
The plane's speed is c=2 and the wind velocity is w=1 due north.  
Find the arc-length of this flight path.  

Example 8.  Use the flight path  {x[t],y[t]}  starting at the point is {1,0}  in example 2.
The plane's speed is c=1 and the wind velocity is w=1 due north.
Find the arc-length of this flight path.  
Why doesn't the plane fly into quadrant II ?

Example 9.  Use the flight path  {x[t],y[t]}  starting at the point is {1,0}  in example 3.
The plane's speed is and the wind velocity is w=1 due north.
Find the arc-length of this flight path.  
What is the plane's limiting velocity ?