An experimenter measures the length of a rod. In the scenarios...

The measured length of a rod is given by $L = L_0 \sqrt{1 - \dfrac{u^2}{c^2}}$, where $L_0$ is the proper length and $u$ is the relative speed between the rod and the observer.

The measured length is minimum when the relative speed $u$ is maximum.

In scenario (1), the relative speed is $u = v$.

In scenario (2), the relative speed is $u = 0$.

In scenario (3), the relative speed is $u = v$.

In scenario (4), the rod and the experimenter move in opposite directions. The relative speed is given by the relativistic velocity addition formula:

$u = \dfrac{v + v}{1 + \dfrac{v^2}{c^2}} = \dfrac{2v}{1 + \dfrac{v^2}{c^2}}$

Since $u > v$ for any $v > 0$, the relative speed is maximum in this scenario.

Thus, the measured length is minimum when the rod and the experimenter travel in opposite directions at the same speed $v$.