Correct Answer - Option 3 : 0.8c

__Concept:__

If we measure the length of anything moving relative to our frame, we find its length L to be smaller than the proper length L0 that would be measured if the object were stationary.

At relativistic speeds, Close to the speed of light, distances measured are not the same when measured by different observers.

__Length Contraction__:

Length contraction is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object.

It is given by

\(L = {L_0}\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \)

where L0 is the length of the object in its rest frame, and L is the length in the frame moving with velocity v;

__Calculation:__

Given:

L0 = 5 m and L = 3 cm

Using, \(L = {L_0}\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \)

⇒ \(3 = 5\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} \)

⇒ \(1-\frac{9}{{25}} =\frac{{{v^2}}}{{{c^2}}}\)

⇒ \(\frac{{{v^2}}}{{{c^2}}} =\frac{{16}}{{25}} \)

\(\frac{v}{c}=\frac{4}{5}\)

v = 0.8c