Question

An experimenter measures the length of a rod. In the cases listed, all motions are with respect to the lab and parallel to the length of the rod. In which of the cases the measured length will be minimum?

(a) The rod and the experimenter move with the same speed v in the same direction.
(b) The rod and the experimenter move with the same speed v in opposite directions.
(c) The rod moves at speed v but the experimenter stays at rest.
(d) The rod stays at rest but the experimenter moves with the speed v.

Answer 1

(b) The rod and the experimenter move with the same speed v in opposite directions.

If a rod is moving with speed v parallel to its length l_o and the experimenter is at rest, its new length will be given as,
$l=l_o\sqrt{1-\frac{v^2}{c^2}}$
If the rod is at rest and the observer is moving with speed v parallel to measured length of the rod, the rod's length will be given as,
$l=l_o\sqrt{1-\frac{(-v{)}^2}{c^2}}=l_o\sqrt{1-\frac{v^2}{c^2}}$
If the rod and the experimenter both are moving with the same speed in the same direction, then l = l_​o while if they are moving with same speed in the opposite directions, the length of the rod will be given as,
$l=l_o\sqrt{1-\frac{(v-(-v){)}^2}{c^2}}=l_o\sqrt{1-\frac{4v^2}{c^2}}$
Where, v<c​
$$\begin{gathered} \mathrm{As},1-\frac{4v^2}{c^2}<1-\frac{v^2}{c^2} \\ \therefore l_o\sqrt{1-\frac{4v^2}{c^2}}<l_o\sqrt{1-\frac{v^2}{c^2}}<l_0 \end{gathered}$$
Therefore, the length will be minimum in the case when both are travelling in opposite direction.​