The relation between position $(x)$ and time $(t)$ is given below for a particle moving along a straight line. Which of the following equations represents uniformly accelerated motion? [where $\alpha$ and $\beta$ are positive constants]

$A$ 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.

What kind of motion is represented by the equation $\Delta S = v \Delta t$?

$A$ body starts from the origin and moves along the $X$-axis such that the velocity at any instant is given by $v = (4t^3 - 2t)$,where $t$ is in $s$ and velocity is in $m/s$. What is the acceleration of the particle when it is $2\, m$ from the origin? (in $m/s^2$)

$A$ bullet from a gun is fired into a rectangular wooden block with velocity $u$. When the bullet travels $24\,cm$ through the block along its length horizontally,the velocity of the bullet becomes $\frac{u}{3}$. It then further penetrates into the block in the same direction before coming to rest exactly at the other end of the block. The total length of the block is $........\,cm$.

$A$ particle is projected with velocity $v_0$ along the $x-$axis,and its deceleration is given by $a = -\alpha x^2$. The distance at which the particle stops is