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Q.
A proton and an $\alpha$ - particle start from rest in a uniform electric field. The ratio of times taken by them to travel the same distance in the field is
Acceleration of $\alpha$ - particle,
$a_{r}=\frac{q E}{m}=\frac{2 e E}{4}$
Acceleration of photon, $a_{p}=\frac{e E}{1}$
They travel same distances in time $t_{1}$ and $t_{2}$.
$\Rightarrow \frac{1}{2}\left(\frac{2 e E}{1}\right) t_{1}^{2}=\frac{1}{2}\left(\frac{e E}{1}\right) t_{2}^{2}$
$\Rightarrow \frac{t_{2}^{2}}{t_{1}^{2}}=\frac{1}{2}$
$\Rightarrow \frac{t_{2}}{t_{1}}=\frac{1}{\sqrt{2}}$