A block of mass $M$ is tied to one end of massless rope. The other end of rope is in the hands of a man of mass $2M$ as show in figure. Initially the block and the man are resting on a rough plank of mass $2M$ as shown in figure. The whole system is resting on a smooth horizontal surface. The man pulls the rope. Pulley is massless and frictionless. What is the magnitude of displacement of the plank when the block meets the pulley ......... $m $ (Man does not leave his position on the plank during the pull).

Imagine the situation in which the given arrangement is placed inside a trolley that can move only in the horizontal direction, as shown in figure. If the trolley is accelerated horizontally along the positive $x$ -axis with $a_0$, then If $a_{min}$ and $a_{max}$ are the minimum and maximum values of $a_0$ for which the blocks remain stationary with respect to the surface, then identify the correct statements

Two blocks of same mass $(4\ kg)$ are placed according to diagram. Initial velocities of bodies are $4\ m/s$ and $2\ m/s$ and the string is taut. Find the impulse on $4\ kg$ when the string again becomes taut  .......... $N-s$

If acceleration of $A$ is $2\,m / s ^2$ to left and acceleration of $B$ is $1\,m / s ^2$ to left, then acceleration of $C$ is -

In the arrangement shown in figure the ends $P$ and $Q$ of an unstretchable string move downwards with uniform speed $ U$. Pulleys $A$ and $B$ are fixed. Mass $M$ moves upwards with a speed

In the adjoining figure if acceleration of $M$ with respect to ground is $a$, then