$A$ point traversed half of the distance with a velocity $v_0$. The remaining part of the distance was covered with velocity $v_1$ for half the time and with velocity $v_2$ for the other half of the time. The mean velocity of the point averaged over the whole time of motion is

$A$ ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d/2$. Neglecting subsequent motion and air resistance,its velocity $v$ varies with the height $h$ above the ground as:

The engine of a motorcycle can produce a maximum acceleration $5 \, m/s^2$. Its brakes can produce a maximum retardation $10 \, m/s^2$. What is the minimum time in which it can cover a distance of $1.5 \, km$?

$A$ bogie of a uniformly moving train is suddenly detached from the train and stops after covering some distance. What is the relation between the distance covered by the bogie and the distance covered by the train in the same time?

Statement $(I)$: An object subjected to velocities $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ has a resultant velocity with magnitude $|\vec{v}| = |\overrightarrow{v_1}| + |\overrightarrow{v_2}|$.
Statement $(II)$: The magnitude of displacement is either less than or equal to the path length of an object between two points.
Statement $(III)$: The instantaneous acceleration is the limiting value of the average acceleration as the time interval approaches zero.
Which of the following is correct?

The relation between time $t$ and distance $x$ is given by $t = \alpha x^2 + \beta x$,where $\alpha$ and $\beta$ are constants. The retardation of the particle is: