圆形场地逃离拴在围栏上的狗

分两阶段。

阶段 1：先绕圈“控角度”。

你先沿以圆心为中心、半径为 $r$ 的圆周跑动，使自己与狗在圆心处的极角保持一致（让狗始终在与你同一径向方向的围栏点上）。

你在半径 $r$ 处的最大角速度为 $\nu/r$；狗在围栏半径 $R$ 处的角速度为 $4\nu/R$。

只要

$$\frac{\nu}{r}\ge \frac{4\nu}{R}\Rightarrow r\le \frac{R}{4},$$

你就能在角度上不被狗甩开。

取 $r=R/4$，在该圆上绕跑，直至你与狗处于“正对”（同一径向）。

阶段 2：径向冲刺。

此时你沿半径方向直线冲向围栏，距离为 $R-r=3R/4$，所需时间

$$t_\text{you}=\frac{3R/4}{\nu}.$$

狗要从对侧沿围栏跑到你冲刺的出口点，最短也需跑半圈，距离 $\pi R$，时间

$$t_\text{dog}=\frac{\pi R}{4\nu}.$$

因为 $t_\text{you}<t_\text{dog}$ 等价于 $3<\pi$（成立），所以你能在狗赶到前越过围栏。

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英文解析

Proceed in two stages.

Stage 1: Run in a circle to control the angle.

First, run along a circular path centered at the origin with radius $r$, adjusting your position so that your polar angle relative to the center matches the dog's (keeping the dog always on the same radial line as you on the fence).

Your maximum angular velocity at radius $r$ is $\nu/r$; the dog's angular velocity at the fence radius $R$is $4\nu/R$.

As long as

$$\frac{\nu}{r}\ge \frac{4\nu}{R}\Rightarrow r\le \frac{R}{4},$$

you will not be overtaken by the dog in terms of angle.

Set $r=R/4$ and run along this circle until you are "opposite" to the dog (on the same radial line).

Stage 2: Radial sprint.

At this point, sprint radially in a straight line toward the fence, covering a distance of $R-r=3R/4$. The time required is

$$t_\text{you}=\frac{3R/4}{\nu}.$$

The dog must run along the fence from the opposite side to your exit point; the shortest distance is a half-circle, $\pi R$, taking time

$$t_\text{dog}=\frac{\pi R}{4\nu}.$$

Since $t_\text{you}<t_\text{dog}$ is equivalent to $3<\pi$ (which holds), you can cross the fence before the dog arrives.