A passenger is standing on the platform at the beginning of the $n^{th}\left(=3^{rd}\right)$ coach of a train. The train starts moving with constant acceleration. The third coach passes by the passenger in $∆t_1=5.0 \mathrm{s}$ and rest of the train (including the $3^{rd}$ coach) in $∆t_2=20 \mathrm{s}$. In what time interval $∆t$ (in second) will the last coach pass by the passenger?

(Take $\sqrt{15}=3.8$)

Two skaters, $P$ and $Q$ are skating towards each other. Skater, $P$ throws a ball towards, $Q$ every $5 \mathrm{s}$ such that it always leaves her hand with speed $2 \mathrm{m} {\mathrm{s}}^{-1}$ with respect to the ground. Consider two cases: (I) $P$ runs with speed $1 \mathrm{m} {\mathrm{s}}^{-1}$ towards $Q,$ while $Q$ remains stationary. (II) $Q$ runs with speed $1 \mathrm{m} {\mathrm{s}}^{-1}$ towards $P,$ while $P$ remains stationary. Note That irrespective of speed of $P,$ ball always leaves $P$ s hand with speed $2 \mathrm{m} {\mathrm{s}}^{-1}$ with respect to the ground. Ignore gravity. Balls will be received by $Q.$

A particle starts from origin $\mathrm{O}$ from rest and moves with a uniform acceleration along the positive $x-\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{s}$ . Identify all figures that correctly represent the motion qualitatively.
( $\mathrm{a}=$ acceleration, $v=$ velocity, $x=$ dispalcement, $\mathrm{t}=$ time)
$\left(\mathrm{A}\right)$
$\left(\mathrm{B}\right)$
$\left(\mathrm{C}\right)$
$\left(\mathrm{D}\right)$

In the cube of side $a$ shown in the figure, the vector from the central point of the face $ABOD$ to the central point of the face $BEFO$ will be: