$A$ block of mass $M$ placed inside a box descends vertically with acceleration $a$. The block exerts a force equal to one-fourth of its weight on the floor of the box. The value of $a$ will be .............

$A$ block of mass $m$ moving with a velocity $v_0$ on a smooth horizontal surface strikes and compresses a spring of stiffness $k$ until the mass comes to rest,as shown in the figure. This phenomenon is observed by two observers:
$A$: standing on the horizontal surface
$B$: standing on the block
To an observer $B$,when the block is compressing the spring:

$A$ person in an elevator accelerating upwards with an acceleration of $2\,ms^{-2}$,tosses a coin vertically upwards with a speed of $20\,ms^{-1}$. After how much time will the coin fall back into his hand? (Take $g = 10\,ms^{-2}$)

$A$ pendulum of mass $m$ hangs from a support fixed to a trolley. The direction of the string when the trolley rolls up an inclined plane of inclination $\alpha$ with acceleration $a_0$ is (the string and bob remain fixed with respect to the trolley):

$A$ wire is bent into the shape of a parabola given by $Y = Kx^2$. $A$ bead of mass $m$ is placed on the wire,which can slide on it without friction. Initially,when the wire is at rest,the bead is at the lowest point of the parabola. If the wire is now moved with a constant acceleration $a$ parallel to the $X$-axis,what will be the distance of the new equilibrium position of the bead from the $Y$-axis,such that it remains stationary relative to the wire?

$A$ container of mass $m$ is pulled by a constant force $F$. Inside the container,a second block of the same mass $m$ is placed,connected to the wall by a massless spring of constant $k$. Initially,the spring is at its natural length. Find the velocity of the container at the instant the compression in the spring is maximum for the first time.