= Solution
Let $e>0$ be the magnitude of the <electron>'s <electric charge>, $m=m_e$, $F=eE>0$, and $\ell=mc^2/F$. The <Lorentz force> is $dp_x/dt=qE$, so the <electron> and <Positron> acquire opposite <momenta>:
$$
p_{e,x}=-Ft,\qquad p_{p,x}=Ft,\qquad
\mathcal E_e=\mathcal E_p=\sqrt{m^2c^4+F^2c^2t^2}.
$$
Using the <relativistic energy-momentum relation>, $u_x=pc^2/\mathcal E$, and integrating from rest gives the <world lines>
$$
\boxed{x_e(t)=-\ell\left[\sqrt{1+(ct/\ell)^2}-1\right],\qquad
x_p(t)=d+\ell\left[\sqrt{1+(ct/\ell)^2}-1\right]\quad(t\geq0).}
$$
For $t<0$, the <world lines> are vertical at $x=0,d$ on a <spacetime diagram>. They join smoothly to the accelerated branches with zero initial <velocity>. At small $t$, $x_e\simeq-Ft^2/(2m)$ and $x_p-d\simeq Ft^2/(2m)$; at large $t$, their <velocities> approach $-c,+c$, respectively. Each accelerated <world line> is a branch of a <hyperbola>, since $(x_e-\ell)^2-c^2t^2=\ell^2$ and $(x_p-d+\ell)^2-c^2t^2=\ell^2$. The <electric field> separates the opposite <electric charges> in this orientation. The asymptotes are lightlike, while every finite-time tangent is timelike.
\Image[/past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2021-accelerating-pair-worldlines.png]
{title=Accelerated pair in the laboratory and a moving frame}
{description=Original <spacetime diagrams> for $d=\ell$ and $v=0.6c$. Dots mark the two switching events. The dashed line is the laboratory's simultaneous switching surface, which is tilted in the moving <inertial frame>.}
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