$A$ small block slides down on a smooth inclined plane,starting from rest at time $t=0$. Let $S_{n}$ be the distance travelled by the block in the interval $t=n-1$ to $t=n$. Then,the ratio $\frac{S_{n}}{S_{n+1}}$ is

$A$ body initially at rest is moving with uniform acceleration $a$. Its velocity after $n$ seconds is $v$. The displacement of the body in the last $2 \ s$ is

$A$ particle starts moving along a line from zero initial velocity and comes to rest after moving a distance $d$. During its motion,it has a constant acceleration $f$ over $2/3$ of the distance and covers the rest of the distance with constant retardation. The total time taken to cover the distance is

Consider a particle moving along the positive direction of the $X$-axis. The velocity of the particle is given by $v = \alpha \sqrt{x}$ (where $\alpha$ is a positive constant). At time $t = 0$,the particle is located at $x = 0$. Find the time dependence of the velocity and the acceleration of the particle,respectively.

$A$ vehicle starts from rest and accelerates along a straight path at $2 \,m/s^2$. At the starting point of the vehicle, there is a stationary electric siren. How far has the vehicle nearly gone when the driver hears the siren at $94 \%$ of its original frequency when the vehicle was at rest (in $\,m$)? (Speed of sound $= 330 \,m/s$)

$A$ body starts from rest and moves with uniform acceleration. If the distance travelled by it in the first $2 \,s$ is $x_1$ and in the next $2 \,s$ is $x_2$, then $x_1$ and $x_2$ are related as