Explanation

(d) It is clear from figure that the displacement vector $\mathrm{\Delta}\overrightarrow{r}$ between particles $p_{1}$ and $p_{2}$ is $\mathrm{\Delta}\overrightarrow{r} = \overrightarrow{r_{2}} - \overrightarrow{r_{1}} = - 8\widehat{i} - 8\widehat{j}$
$\left| \mathrm{\Delta}\overrightarrow{r} \right| = \sqrt{( - 8)^{2} + ( - 8)^{2}} = 8\sqrt{2}$ ….(i)
Now, as the particles are moving in same direction
$(\because\overrightarrow{v_{1}}$ and $\overrightarrow{v_{2}}$ are $+ ve)$, the relative velocity is given by
${\overrightarrow{v}}_{rel} = \overrightarrow{v_{2}} - \overrightarrow{v_{1}} = (\alpha - 4)\widehat{i} + 4\widehat{j}$
$\left| {\overrightarrow{v}}_{rel} \right| = \sqrt{(\alpha - 4)^{2} + 16}$ …(ii)
Now, we know $\left| {\overrightarrow{v}}_{rel} \right| = \frac{\left| \mathrm{\Delta}\overrightarrow{r} \right|}{t}$
Substituting the values of ${\overrightarrow{v}}_{rel}$ and $\left| \mathrm{\Delta}\overrightarrow{r} \right|$ from equation (i) and (ii) and $t = 2s$, then on solving we get $\alpha = 8$