Calculate the acceleration of the block $B$ in the figure,assuming the surfaces and the pulleys $P_1$ and $P_2$ are all smooth and the pulleys and string are massless.

$A$ frictionless cart $A$ of mass $100 \ kg$ carries two other frictionless carts $B$ and $C$ having masses $8 \ kg$ and $4 \ kg$ respectively,connected by a string passing over a pulley as shown in the figure. What horizontal force $F$ must be applied on the cart $A$ so that the smaller carts do not move relative to it (in $N$)? (Take $g = 10 \ m/s^2$)

$A$ block is dragged on a smooth plane with the help of a rope which moves with a velocity $v$ as shown in the figure. The horizontal velocity of the block is

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

In the arrangement shown in the figure,the block of mass $m=2\,kg$ lies on the wedge of mass $M=8\,kg$. Find the initial acceleration of the wedge if the surfaces are smooth and the pulley and strings are massless.

In each of the three arrangements,the block of mass $m_1$ is being pulled left with constant velocity $v$. There is no friction anywhere. The strings are light and inextensible and pulleys are massless. The ratio of the speed of the block of mass $m_2$ in the three cases respectively is