$A$ car moves in the positive $Y$-direction with velocity $v$ proportional to the distance travelled $y$ as $v(y) \propto y^\beta$, where $\beta$ is a positive constant. The car covers a distance $L$ with average velocity $\langle v \rangle$ proportional to $L$ as $\langle v \rangle \propto L^{1/3}$. The constant $\beta$ is given as:

$A$ metro train starts from rest and in $5 \, s$ achieves $108 \, km/h$. After that,it moves with constant velocity and comes to rest after travelling $45 \, m$ with uniform retardation. If the total distance travelled is $395 \, m$,then the total time of travelling is ....... $s$.

Which of the following curves possibly represent one-dimensional motion of a particle? Choose the correct answer from the options given below.

$A$ particle has an initial velocity of $10 \, m/s$. It moves due to a constant retarding force along the line of motion,which produces a retardation of $5 \, m/s^2$. Then:

$A$ person wearing a parachute jumps off a plane from a height of $2 \ km$ from the ground and falls freely for $20 \ m$ before his parachute opens. After his parachute opens, if he continues to move uniformly with the velocity attained due to his freefall, the total time taken by the person to reach the ground is (Acceleration due to gravity $= 10 \ m/s^2$) (in $s$)

For the velocity-time graph shown in the figure,in a time interval from $t=0$ to $t=6\,s$,match the following columns.

Column $I$	Column $II$
$(A)$ Change in velocity	$(p)$ $-5/3\,SI \text{ unit}$
$(B)$ Average acceleration	$(q)$ $-20\,SI \text{ unit}$
$(C)$ Total displacement	$(r)$ $-10\,SI \text{ unit}$
$(D)$ Acceleration at $t=3\,s$	$(s)$ $-5\,SI \text{ unit}$