$A$ particle located at $x = 0$ at time $t = 0$ starts moving along the positive $X$-direction with a velocity $v$ that varies as $v = \alpha \sqrt{x}$. The displacement of the particle with time is proportional to

$A$ train moves from rest with a uniform acceleration $a$. After attaining a maximum speed $v$,it starts moving with uniform retardation $a$. Assuming $s$ is the total distance covered in the unidirectional motion of the train,find its total time of journey and maximum speed.

If the displacement of a particle varies with time as $\sqrt{x} = t + 7$,then

An object moving along the $x$-axis with a uniform acceleration has a velocity $\vec{v} = (12 \ cm \ s^{-1}) \hat{i}$ at $x = 3 \ cm$. After $2 \ s$,if it is at $x = -5 \ cm$,then its acceleration is:

$A$ rifle bullet loses $\left(\frac{1}{25}\right)^{th}$ of its velocity in passing through a plank. The least number of such planks required just to stop the bullet is

Two cars $A$ and $B$ initially at rest are moving in the same direction with accelerations $a_1$ and $a_2$ respectively. After a certain time $t$, they achieve velocities $v_1$ and $v_2$ respectively and are separated by a distance of $50 \ m$. If $(a_1 - a_2) = 4 \ m \ s^{-2}$, then the quantity $(v_1 - v_2)$ will be: (in $m \ s^{-1}$)