When the speed of a rod travelling parallel to its length at a...

The relativistic mass $m$ of an object moving with velocity $v$ is given by:

$m = \dfrac{m_0}{\sqrt{1 - \dfrac{v^2}{c^2}}}$

The relativistic length $L$ of an object moving with velocity $v$ parallel to its length is given by:

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

When the speed $v$ is doubled to $2v$, the term $\dfrac{v^2}{c^2}$ increases to $\dfrac{4v^2}{c^2}$. Consequently, the factor $\sqrt{1 - \dfrac{v^2}{c^2}}$ decreases.

As the denominator in the mass expression decreases, the relativistic mass $m$ increases.

As the factor in the length expression decreases, the relativistic length $L$ decreases.

Since the dependence on $v$ is not linear, the mass does not exactly double and the length does not exactly become half for an arbitrary initial velocity.