(Corrected version:) Can Haley escape a horrible fate?

Question

This is the corrected version to a puzzle submitted earier. The idea I have in mind should work this time....


Haley is in her circular swimming pool when who should show up but Lewd Larry, the obnoxious guy who has tried fondling her on several occasions. She knows that if she can get to a point on the edge of the pool before Larry does, she can give him a swift punch on the crotch and escape without getting felt up. She also knows she can move 4mph in the pool while Larry can run 14mph around the outside of the pool. Does Haley have a way to escape from the pool unscathed, or is she doomed to a grope-fest from Larry?

Sorry, no fast breaks. Let's assume Haley is in the middle of the pool, while Larry is already at the edge, waiting and salivating...

Alexander must definitely be a physicist, with all this talk of "maximum angular velocity". :) We're coming up with the same strategy, however, just different ways to describe it. My thinking is, Haley can do little circles faster than Larry can do the big circle around the edge of the pool. Given the ratio of Haley's speed to Larry's speed, she can circle faster up to the point her circle is 2/7 the size of Larry's circle.

To make this easier to visualize, let's say the radius of the pool is 14 feet. (The actual size of the pool is irrelevant, of course.) Then, by spiraling away from the center, Haley can get 14 * 2/7 = 4 feet closer to the side and still have Larry the maximum distance from her. At this point, as Alexander says, Haley can make a break for the edge and beat Larry by a comfortable margin.

By my calculation, the critical ratio is π + 1 : 1, about 4.14. If Larry's speed is less than 4.14 times faster than Haley's speed, Haley can escape.

Answer 1

v = 4 mph
V = 14 mph
R pool radius

Haley can swim along the circle radius of radius r
with maximum angular velocity ω = v / r.
Larry can run with angular velocity Ω = V / R.

Inside the circle of radius ρ = vR/V = 2/7 R
Haley can have higher angular velocity and can
choose their angular separation θ as she wishes.
In her place I would choose θ = π, r = ρ,
and swim to the nearest shore.

The distance to be covered by Haley,
d = R - ρ = R(1 - v/V), time
t = d/v = R(1/v - 1/V).

The distance to be covered by Larry,
D = πR, time
T = D/V = πR/V

Condition of escape:
D > d
D - d = πR/V - R(1/v - 1/V) = R((π+1)/V - 1/v) =
R/V (π + 1 - V/v)

2π - V/v = π - 2.5 > 0

Success by wide margin.

Answer 2

Haley can probably escape if she always swims in the direction away from Lewd Larry, which will take her on curved path. She can either immediately leave the pool if she's close enough to the edge, or Lewd Larry will force her back towards the center, after which she'll begin on a spiral-like path towards the edge. it does critically depend on the ratio of their velocities for her to be able to escape, since she has to reach the edge before Larry Lewd can reach the point colinear with Haley and the center, beause once this happens, she'll swim endlessly the pool.

I need to find the time to work out the actual trajectory, I don't know when that will be.

Answer 3

Since 14 > 4π, the answer is that it depends on where she is in the pool and which direction Larry approaches from. If she happens to be near the part of the edge he arrives at, he will be able to go faster around the pool than she can across it, so she's stuck. If she's near the middle or near the farther edge, she can strike out in the direction away from Larry and get to the other side before he can make up the extra distance.