# Lifting a survivor into a helicopter with a rope

## 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##.

Last edited:

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.Scott
Homework Helper
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.

• berkeman
haruspex
Homework Helper
Gold Member
I assume we are to find θ in terms of Fg, v, vk and... something else, but what? Maybe the original FAIR, the original angle, or the coefficient of drag?
Using the original FAIR 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 vk.

... 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?