= Solution
The laboratory remains the <center-of-momentum frame>. At the <relativistic collision mass threshold>, the two final <W bosons> can have arbitrarily small opposite <momenta>. The available <relativistic energy> is $2m_ec^2+eEd$, so
$$
2m_ec^2+eEd\geq2m_{\mathrm W}c^2,\qquad
\boxed{d_{\min}=\frac{2(m_{\mathrm W}-m_e)c^2}{eE}
=\frac{160.756978\times10^9\,\mathrm{eV}}{5.0\times10^6\,\mathrm{eV\,m^{-1}}}
\simeq3.21514\times10^4\,\mathrm m=32.15\,\mathrm{km}.}
$$
Each incoming particle must acquire about $80.378489\,\mathrm{GeV}$ of <kinetic energy>, travelling about $16.08\,\mathrm{km}$. Equality is the ideal kinematic threshold; a nonzero production rate needs <energy> above it. This estimate neglects acceleration losses and treats the <W bosons> with the supplied nominal <rest mass>.
A <linear particle accelerator> avoids repeatedly bending high-<energy> <electrons> and <Positrons>. This greatly reduces the bending <synchrotron radiation> loss, which at fixed bending radius grows approximately as the fourth power of the <relativistic energy> for a given particle species. Its useful collision <energy> can therefore be increased without the severe circular <electron> collider radiation penalty. The price is long structures or high accelerating gradients: each bunch normally passes once, requiring substantial beam power, fresh bunches, and very tight focusing for high <luminosity>. A circular <particle accelerator> reuses its accelerating structures and circulates bunches through many collisions, but needs a large ring and substantial radiofrequency power to replace <synchrotron radiation> losses. Heavier particles such as <protons> radiate far less at a given <energy> and radius, although a <proton>'s constituent structure makes the hard-collision <energy> less sharply specified than in an <electron-positron annihilation> experiment.
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