Finding the Optimal Flight Path to the Origin: A Vector Addition Problem

[Altheides]

Homework Statement

A pilot attempts to y with constant speed v from the point P = (D; 0) on the x-axis to the
origin O = (0; 0). A wind blows with speed w in the positive y direction. The pilot is not familiar
with vector addition and thinks the shortest path to O is achieved by
ying his plane so that it always points directly towards O.
(a) Show that the actual flight path of the plane (in Cartesian coordinates) is given by
y(x) = f(x) sinh[g(x)];
where f(x) and g(x) are scalar functions of x that are to be determined.

Homework Equations

No formula's given but we have been working with tangential and normal acceleration components

The Attempt at a Solution

The plane always points towards to origin. So I would assume the force of the engines is directed towards the origin. This I would assume the solution to be

a(t) = -m(t)*x(t)
or a(x) = -n(x) * (x,y(x))

I am not sure how to handle the wind. It would seem out of place to do something with draf forces given the content of the course but I am not really that into physics so who knows. It may just be an initial velocity. Or that everything the pilot does is relative to the moving air.

I also though of this expression for the velocity

v(t) = -v*x(t)/|x(t)| +w*jhat

That I can actually treat as a system of linear equations but the solution seems complicated too complicated for this course.

My thought is that once I get the right differential equation I just show that the proposed form in the question satisfies that differential equation.

Please Help. Thanks in advance.

 

[Altheides]

There is a part c to this problem as well that I am having trouble with.

(c)
Show that the flight path of the plane (in polar coordinates) is given by the polar equation

r = r(theta) = D(cos(theta))^Beta1(1 + sin(theta))^Beta2;

where Beta1 and Beta2 are constants (that depend on v and w) that are to be determined.

I have tried solving this but the resulting DE doesn't really match.

I get that
d(theta)/dt = w*cos(theta)
dr/dt = -v + w*sin(theta)

=> dr/d(theta) = (sin(theta) -v/w)/cos(theta)

but the solution to this is complicated maybe it matches the form from the question but I can't determine that.

Please any! help here would be most appreciated. I'm getting pretty frustrated with the question.

 

[ehild]

The plane moves with constant speed v with respect to the wind, and the direction of its velocity (not the acceleration) points towards the origin in the frame of reference connected to the wind. The wind blows in the y direction with speed W. The velocity of the plane in the inertial frame of reference is the sum of v, the velocity of the plane with respect to the wind, plus the velocity of the wind W. Write out the x, y components of the velocity in the inertial frame of reference.

ehild