(a)(i) An electron of charge $-e$ in field $E$ experiences force $F=eE$ and acceleration $a=\dfrac{eE}{m}$.

Between two successive collisions (average time = relaxation time $\tau$), the electron gains an additional velocity $a\tau$ on average (starting each time from a randomised velocity after collision).

Averaging over many electrons, the drift velocity:

$$v_d = a\tau = \frac{eE\tau}{m}$$

Since $E=V/L$ (potential $V$ across length $L$): $v_d=\dfrac{eV\tau}{mL}$

(ii) Although each electron is continuously accelerated by the field between collisions, it undergoes frequent, random collisions with the lattice ions (relaxation time $\tau\sim10^{-14}$ s is extremely small). Each collision randomises the electron's velocity, effectively 'resetting' the velocity gained from the field. The NET effect, averaged over a large number of electrons and collisions, is a small constant average drift velocity superimposed on the much larger random thermal velocities — rather than an ever-increasing (accelerating) velocity.