Prisoner Puzzle (Mathematical)

This is a "Prisoner and Gaoler/Jailer" problem.

The primary answer required is a number, correct to 4 significant decimal digits.

There are 3 plausible answers of which only 1 is correct.

It is expected that the solution will require:

• A writing implement (pen/pencil)
• 1 Sheet of A4 paper: diagrams on 1 side, workings on the other
• An electronic calculator (or equivalent)
• Basic (school level) skills in:
  • Arithmetic
  • Algebra
  • Geometry
  • Trigonometry
  • Differential Calculus
  • Integral Calculus

The prison consists of a circular lake, 1 unit of distance in radius. (Miles may be convenient.)

The lake is stocked with piranhas/ alligators/ great white sharks/ income tax inspectors or lawyers as may suit your imagination.

The prisoner is initially located at the centre of the lake and is supplied with a rowing boat for exercise purposes.

The prisoner can row, row, row her boat at a maximum of 1 unit of speed. (Miles per hour may be convenient.)

The gaoler/jailer is a goddamn' retiree suffering from arthritis, rheumatism, etc, and is supplied with a mobility scooter for patrol purposes.

The mobility scooter has a maximum speed and starts on the lake shore, but cannot manoeuvre within the lake.

The prison governor requires to know: How fast does the mobility scooter have to be to enable the gaoler/jailer to keep the prisoner trapped within the lake?

It may be assumed for the purposes of calculation that:

• The world is flat. (No corrections for spherical geometry are required.)
• The prisoner, boat, gaoler/jailer and mobility scooter are arbitrarily small.
• The prisoner+boat and the gaoler/jailer+mobility scooter can each accelerate arbitrarily quickly (up to their respective maximum speeds).
• The prisoner and gaoler/jailer have good eyesight and each can readily see across the lake and assess accurately what the other is doing.
• The prisoner and the mobility scooter have sufficient stamina to complete all required manoeuvres.
• The prisoner and gaoler/jailer are sufficiently intelligent to identify at any time their most appropriate tactics.

Additionally required is a good, mathematical description of the paths and times taken by the prisoner and gaoler/jailer during a critical escape attempt.

To give compatible diagrams, where the prisoner or gaoler/jailer has a free symmetric choice of turning off a straight-line course, please assume that each alters anti-clockwise.

By consulting a plausible escape diagram, determine whether it is better to use the spelling "Gaoler" or "Jailer".