Lifting a survivor into a helicopter with a rope

[bolzano95]

Homework Statement

A helicopter is flying horizontally with constant velocity $v$ and carries a survivor who hanging on a rope with length $l$. Suddenly the pilot receives a message he has to pull the the survivor into the the helicopter as soon as possible. A paramedic on board starts to pull a hanging rope with such force that the length of rope is shortening with velocity $v_k$.

What is the angle $\theta$ between the rope and vertical line?

Relevant Equations

Relevant equations are below.

We have 2 forces affecting the rope: 1. Gravitational force of the body $=mg$ and 2. Force of air = Force of drag= $F_{AIR}$.
The length of the rope is shortening with the velocity $v_k$.
So to figure out the angle $\theta$ I wrote:
$R$= force of rope
$R_x = F_{AIR}$
$R_y = mg$

Now I know the acceleration is the first derivative of velocity, but don't know how to incorporate this with the angle $\theta$, length of the rope and $v_k$.

 

[.Scott]

Acceleration is not an issue. The reason that pulling the rope up matters is that is changes the velocity of the body through the air. There will be a downward component to the drag on the hanging body.

So $F_{AIR}$ will not be exactly horizontal.

 

[haruspex]

I assume we are to find θ in terms of F_g, v, v_k and... something else, but what? Maybe the original F_AIR, the original angle, or the coefficient of drag?
Using the original F_AIR I get a quartic in tan(θ). The other choices I listed will be similar.
Perhaps you are expected to make an approximation for smallish v_k.

 

[Lnewqban]

... What is the angle $\theta$ between the rope and vertical line?
Relevant Equations:: Relevant equations are below.

What are those relevant equations that are mentioned in the OP?