$A$ 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 varies with time as:

If a train travelling at $72 \text{ km/h}$ is to be brought to rest in a distance of $200 \text{ m}$,then its retardation should be ............ $\text{m/s}^2$.

$A$ bullet enters a wooden block with velocity $120 \,m/s$. The bullet travels $1.5 \,s$ in the block before its velocity reduces to zero due to a resistance force which is proportional to the square root of the velocity. The distance travelled by the bullet in the wooden block is (in $\,m$)

The acceleration $a$ in $m/s^2$ of a particle is given by $a = 3t^2 + 2t + 2$,where $t$ is the time. If the particle starts with an initial velocity $u = 2\,m/s$ at $t = 0$,then the velocity at the end of $2\,s$ is:

$A$ particle initially at rest moves along the $x$-axis. Its acceleration varies with time as $a = 4t$. If it starts from the origin,the distance covered by it in $3 \ s$ is $........... \ m$.

The velocity of an object is given by $v = kt$,where $k = 2 \, m/s^2$. What distance (in $m$) will it cover in the first $3 \, s$?