= Solution
The <relativistic kinetic energy> is the total <energy> minus the <rest energy>:
$$
\boxed{K=(\gamma-1)Mc^2,\qquad\gamma=\frac1{\sqrt{1-v^2/c^2}}.}
$$
Here $\gamma$ is the <Lorentz factor> and $c$ the <speed of light>. The <binomial series>, with $b=v/c$, gives
$$
(1-b^2)^{-1/2}=1+\frac12b^2+\frac38b^4+O(b^6),\qquad K=\frac12Mv^2+\frac38M\frac{v^4}{c^2}+O\!\left(M\frac{v^6}{c^4}\right).
$$
For $v\ll c$, the leading term is the classical <kinetic energy> $Mv^2/2$. The ratio of the fourth-order term to that leading term is $3v^2/(4c^2)$. Consequently
$$
\frac34\frac{v^2}{c^2}=0.01\quad\Longrightarrow\quad\boxed{\frac vc=\sqrt{\frac{0.04}{3}}=0.11547\simeq11.5\%.}
$$
The condition concerns one term of the <binomial series>; it is not the condition that the entire relativistic correction be exactly $1\%$.
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