# Aircraft Guidance in a Crosswind

1. Nov 17, 2007

### bigred

This is a differential equations problem that I've been having some trouble with. A plane is flying and always pointing towards point (0,0) on an xy plane. Wind is constantly blowing North (positive y direction). The wind speed and speed of the aircraft through the air are constant.

(a) Locate the flight in the xy-plane, placing the start of the trip at (2,0) and the destination at (0,0). Set up a differential equation describing the aircrafts path over the ground.

(b) Make an appropriate substitution and solve this equation.

(c) Use the fact that x = 2 and y = 0 at t = 0 to determine the appropriate value of the arbitrary constant in the solution set.

(d) Solve to get y explicitly in terms of x. Write your solution in terms of a hyperbolic function.

(e) Let gamma be the ratio of windspeed to airspeed. Using a software package, graph solutions for the cases gamma = 0.1, 0.3, 0.5 and 0.7 all on the same set of axes. Interpret these graphs.

(f) Discuss the (terrifying!) cases gamma = 1 and gamma greater then 1.

My attempts to find a solution to this word problem are pretty pathetic. Any help would be appreciated.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 17, 2007

### dynamicsolo

Your differential equation is going to be based on the fact that the aircraft's velocity over the ground at any moment is the sum of two velocities:

the wind speed, which has a constant direction (as well as speed), and

the aircraft's velocity relative to the air, which has a constant speed but a direction pointing radially toward the origin .

So you'll need to work out how to express the radial direction. It will probably be easier to work in Cartesian coordinates. You will also probably want to start with two differential equations, one for dx/dt and another for dy/dt, and then construct a single equation from those. It will also make the later questions easier to deal with by calling the aircraft's speed v and the windspeed (gamma)·v .