An airplane and the wind
A little airplane tries to fly from A to B, which is not too far from
A (say 100 miles) and is exactly to the East of A. The wind is blowing
from the South and the speed of the wind is exactly equal to the speed
of the airplane. (The speed of the airplane is measured
with respect to the air!) The pilot decides to steer straight to B all
the time during the flight. Will the airplane ever reach B?
What if the speed of the wind is k times the speed of the airplane,
where k is a positive number (can be greater or less than 1)?
Try to sketch the trajectory of the airplane (with respect to the ground,
of course) in each of the three cases: k=1, k>1 and k<1.
Can you solve the same problem if the wind blows from SW, or more
generally from some arbitrary fixed direction?
Do you see any similarity with the `Little Jo and her pig' problem?
Maybe after you solve both?
Hints and comments.
The distance (100 miles) is of course
irrelevant. It was given only as a hint that the Earth can be considered flat.
Introduce the coordinate system with A at the origin, B=(1,0) and the y-axis
pointing North. Let y=f(x) be the trajectory of the airplane.
At every point (X,Y) on this trajectory the actual velocity of the airplane
(with respect to the ground) is a vector sum of two vectors of equal length:
one pointing North, another from (X,Y) to B. This permits you to write a
differential
equation, and to find f(x) by solving this equation with appropriate
initial conditions.