= Solution
Express <temperatures> as energies $\Theta=k_BT$. A longitudinal cathode-energy spread of order $\Theta_c$ remains of that order during an ideal common electrostatic <acceleration>. At final mean <speed> $v_0$, a small <energy> spread produces $\delta v_\parallel\simeq\delta E/(m_ev_0)$. In the accompanying beam frame, the corresponding random kinetic-energy scale is
$$
\Theta_\parallel\sim\frac{m_e\delta v_\parallel^2}{2}
\sim\frac{\Theta_c^2}{2m_ev_0^2}
=\boxed{\frac{\Theta_c^2}{4E_e}}.
$$
The last equality uses the nonrelativistic <acceleration> <energy> $E_e=m_ev_0^2/2$. This is the approximate <temperature> convention supplied in the hint; exact numerical factors depend on the emitted distribution and the definition of spread. At $50$ keV, relativistic corrections are modest but not strictly zero.
Electrostatic <acceleration> narrows longitudinal <velocity> spread relative to the fast mean beam even though it does not remove each <electron>'s transverse cathode motion. In ideal magnetized cooling, slow <proton> motion couples to the <electron> guiding-centre dynamics, making this very small longitudinal spread relevant to the attainable <ion> <temperature>. Practical voltage ripple, space charge, alignment errors and <ion> heating can set much larger limits.
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