= Solution
Write $E=|\mathbf E|$ and $\widehat{\mathbf E}=\mathbf E/E$. The constant electric <force> gives <momentum>
$$
\mathbf p(t)=q\mathbf E t.
$$
Combining $\mathbf p=\gamma m\dot{\mathbf x}$ with the <relativistic energy-momentum relation> gives
$$
\boxed{\dot{\mathbf x}(t)=\frac{q\mathbf E t/m}{\sqrt{1+(qEt/mc)^2}}}.
$$
The <dimensional analysis> from part (a) confirms that $qEt$ has the dimensions of <momentum>, so $qEt/(mc)$ is dimensionless and the expression has the dimensions of <velocity>.
The <speed> increases monotonically from zero and approaches the <speed of light> asymptotically:
$$
\lim_{t\to\infty}|\dot{\mathbf x}(t)|=c.
$$
Integrating in the direction of the field gives
$$
\boxed{\mathbf x(t)=\frac{mc^2}{qE}\left[\sqrt{1+\left(\frac{qEt}{mc}\right)^2}-1\right]\widehat{\mathbf E}}.
$$
At small times, the <Taylor expansion> of the square root gives the <nonrelativistic limit>
$$
\mathbf x(t)=\frac{q\mathbf E}{2m}t^2+O(t^4),
$$
which is motion from rest with the constant <acceleration> $q\mathbf E/m$.
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