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?

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KollegeApply · Accepted Answer

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 lo and the experimenter is at rest, its new length will be given as, l = l_o $\sqrt$(1 - 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 - ( - v )^2/c^2) = l_o $\sqrt$(1 - 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 - 4 v^2/c^2) Where, v < c As, 1 - 4 v^2/c^2 < 1 - v^2/c^2 \therefore l_o $\sqrt$(1 - 4 v^2/c^2) < l_o $\sqrt$(1 - v^2/c^2) < l_0 Therefore, the length will be minimum in the case when both are travelling in opposite direction.​

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