A plane must fly eastwards from the point (a,0) to the point (0,0). The plane travels at a constant speed v0 relative to the wind, which is blowing northwards with constant speed w. The pilot keeps the plane pointing directly at the destination. What is the plane's trajectory?
This problem is described on page 57 of Edwards & Penney (1996). The horizontal and vertical velocities of the plane are
dx/dt = -v0 x / sqrt (x2 + y2),The trajectory of the plane satisfies the differential equation
dy/dt = -v0 y / sqrt (x2 + y2) + w.
dy/dx = y / x - k sqrt (1 + y / x)were
k = w/v0is the ratio of the wind speed to the plane's air speed. The trajectory can be found by substituting y = v x, giving the solution
y = [a / 2] [(x / a)1-k - (x / a)1+k]
The trajectory passes through the destination (0,0) only when k < 1.
The animation at the top of the page shows the trajectory when k = 0.8. The grid is 100km squares. The speed of the plane is 1000km/h. The destination is the lower left corner of the grid. The animation was generated using a Runge-Kutta scheme to solve dx/dt and dy/dt.
When k = 1 the plane approaches the point (0,a/2).
At this point the wind speed balances the air speed, and the plane is stuck.
When k > 1, the plane disappears somewhere up north. (Where?)
