Q.How fast would a rocket have to go relative to an observer for its length to be corrected to 99% of its length at rest ? ($c = 3\times10^8$ ms$^{-1}$)

Length contraction is the relativistic effect whereby the length of an object, measured by an observer relative to whom it is moving, is found to be smaller than its proper length, by the factor $\sqrt{1-v^2/c^2}$.

Let a rod be at rest in frame $S'$, which moves with velocity $v$ along the $x$-axis relative to frame $S$. Let $L_0 = x_2' - x_1'$ be the proper length of the rod — its length as measured by an observer at rest relative to the rod (in $S'$), where $x_1'$ and $x_2'$ are the (time-independent) positions of its two ends in $S'$.

An observer in frame $S$, relative to whom the rod is moving, must measure the positions $x_1$ and $x_2$ of the two ends of the rod at the same instant of time $t$ (in $S$) to obtain its length $L = x_2 - x_1$ in that frame — measuring the ends at different times would give a meaningless result for a moving object.

Using the Lorentz transformation $x' = \gamma(x - vt)$, where $\gamma = 1/\sqrt{1-v^2/c^2}$, and applying it to both ends at the same time $t$:

$x_1' = \gamma(x_1 - vt), \qquad x_2' = \gamma(x_2 - vt)$

Subtracting,

$L_0 = x_2' - x_1' = \gamma(x_2 - x_1) = \gamma L$

so that

$L = \dfrac{L_0}{\gamma} = L_0\sqrt{1 - \dfrac{v^2}{c^2}}$

Since $\gamma \ge 1$ for any $v>0$, we have $L \le L_0$: the length measured by an observer relative to whom the rod moves is always less than (or equal to) its proper length. This is length contraction (also called the Lorentz–FitzGerald contraction).

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Special relativity is built on two postulates: the equivalence of all inertial frames for the laws of physics, and the constancy of the speed of light in vacuum for every inertial observer.

Background

Einstein put forward the special theory of relativity in 1905 to resolve the conflict between Newtonian mechanics and Maxwell's electromagnetic theory, particularly the failure of experiments (such as the Michelson–Morley experiment) to detect the motion of the Earth through a hypothetical 'ether'. The theory applies to inertial frames of reference — frames that are either at rest or moving with uniform (constant) velocity relative to one another, with no acceleration.

First postulate — the principle of relativity

The laws of physics are the same in all inertial frames of reference. In other words, no experiment performed entirely within a given inertial frame can determine whether that frame is at rest or moving uniformly; there is no absolute or preferred inertial frame, and the equations describing mechanics, electromagnetism, etc. take an identical form for every inertial observer.

Second postulate — constancy of the speed of light

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