A motion sensor sits at the origin, in the middle of a field. The probability that you are detected as you sneak from one point to another along a path $\mathbf{x}(t): 0 \leqslant t \leqslant T$ is

$$P[\mathbf{x}(t)]=\lambda \int_{0}^{T} \frac{v(t)}{r(t)} d t$$

where $\lambda$ is a positive constant, $r(t)$ is your distance to the sensor, and $v(t)$ is your speed. (If $P[\mathbf{x}(t)] \geqslant 1$ for some path then you are detected with certainty.)

You start at point $(x, y)=(A, 0)$, where $A>0$. Your mission is to reach the point $(x, y)=(B \cos \alpha, B \sin \alpha)$, where $B>0$. What path should you take to minimise the chance of detection? Should you tiptoe or should you run?

A new and improved sensor detects you with probability

$$\tilde{P}[\mathbf{x}(t)]=\lambda \int_{0}^{T} \frac{v(t)^{2}}{r(t)} d t$$

Show that the optimal path now satisfies the equation

$$\left(\frac{d r}{d t}\right)^{2}=E r-h^{2}$$

for some constants $E$ and $h$ that you should identify.