Rope swing physics problem-please help explain.

Question

You and your friends find a rope that hangs down 19 m from a high tree branch right at the edge of a river. You find that you can run, grab the rope, swing out over the river, and drop into the water. You run at 2.0 m/s and grab the rope, launching yourself out over the river.

How long must you hang on if you want to drop into the water at the greatest possible distance from the edge?

Answer 1

What a fun question! I think we should run a bunch of experiments to figure it out!

For this first-pass analysis, let's assume:

1. the water level is h_r below your center of mass
2. your mass and geometry can be modeled as a particle

Key facts to recognize:

1. After you release your grip, your center of gravity's flight path is a parabola.
2. The problem has 2 distinct parts or phases:
3. while gripping the rope
4. after release
5. The initial conditions for part b come from the analysis of part a. That is, your lateral distance at impact depends on your launch speed and direction and lateral position at the time you release.
6. And your launch speed and direction (angle) and lateral position depend on the angle of the rope at the time you release.

For part a, consider using conservation of energy to derive expressions for launch speed, direction, and position.

Part b is a ballistics problem; derive an expression for the lateral distance.

Finally, substitute and solve.