The velocity $v$ of a particle as a function of its position $x$ is expressed as $v = \sqrt{c_1 - c_2 x}$,where $c_1$ and $c_2$ are positive constants. The acceleration of the particle is

Displacement $(x)$ of a particle is related to time $(t)$ as: $x = at + bt^2 - ct^3$,where $a, b$,and $c$ are constants of the motion. The velocity of the particle when its acceleration is zero is given by:

$A$ lift is coming down from the $8^{th}$ floor and is just about to reach the $4^{th}$ floor. Taking the ground floor as the origin and the upward direction as positive for all quantities,which one of the following is correct?

Give examples of a one-dimensional motion where:
$(a)$ The particle moving along positive $x$-direction comes to rest periodically and moves forward.
$(b)$ The particle moving along positive $x$-direction comes to rest periodically and moves backward.

$A$ car moving at a speed $V$ is stopped in a certain distance $s$ when the brakes produce a deceleration $a$. If the speed of the car is $nV$,what must be the deceleration of the car to stop it in the same distance $s$ and in the same time $t$?

$A$ particle starts moving from rest in a straight line with constant acceleration. After time $t_0$,the acceleration changes its sign (becomes exactly opposite to the initial direction),while remaining the same in magnitude. Determine the time from the beginning of the motion at which the particle returns to its initial position.