Within a discharge tube filled with a gas, a molecule undergoes...

The force experienced by the electron in the electric field is given by $F = eE$.

The acceleration of the electron is $a = \dfrac{eE}{m}$.

Using the equation of motion $s = ut + \dfrac{1}{2}at^2$ with initial velocity $u = 0$, the distance travelled is:

$s = \dfrac{1}{2} \left( \dfrac{eE}{m} \right) t^2$

Substituting the standard values $e = 1.6 \times 10^{-19} \text{ C}$, $m = 9.1 \times 10^{-31} \text{ kg}$, and the given values $E = 5.0 \times 10^3 \text{ V m}^{-1}$, $t = 10^{-6} \text{ s}$:

$s = \dfrac{1}{2} \times \dfrac{1.6 \times 10^{-19} \times 5.0 \times 10^3}{9.1 \times 10^{-31}} \times (10^{-6})^2$

$s = \dfrac{1}{2} \times \dfrac{8.0 \times 10^{-16}}{9.1 \times 10^{-31}} \times 10^{-12}$

$s = \dfrac{4.0}{9.1} \times 10^3$

$s = \dfrac{4000}{9.1} \approx 439.56 \text{ m}$

Rounding to the nearest given option, the distance travelled is $440 \text{ m}$.