Consider the diagram shown below in which two masses of $m$ and $2m$ are placed on a fixed triangular wedge. The coefficient of friction between block $A$ (mass $m$) and the wedge is $\mu_A = 2/3$,while that for block $B$ (mass $2m$) and the wedge is $\mu_B = 1/3$. Find the acceleration of the system.

$A$ $100 \, kg$ gun fires a ball of $1 \, kg$ horizontally from a cliff of height $500 \, m$. It falls on the ground at a distance of $400 \, m$ from the bottom of the cliff. Find the recoil velocity of the gun. (Acceleration due to gravity $g = 10 \, m/s^2$) (in $, m/s$)

Two pulley arrangements shown in the figure are identical. The mass of the rope is negligible. In figure $(a)$,the mass $m$ is lifted by attaching a mass $2m$ to the other end of the rope. In figure $(b)$,$m$ is lifted up by pulling the other end of the rope with a constant downward force $F = 2mg$. The accelerations of $m$ in the two cases are respectively:

$A$ $300\ kg$ crate is dropped vertically onto a conveyor belt that is moving at $1.20\ m/s$. $A$ motor maintains the belt's constant speed. The belt initially slides under the crate,with a coefficient of friction of $0.400$. After a short time,the crate is moving at the speed of the belt. During the period in which the crate is being accelerated,find the work done by the motor which drives the belt :- ................. $J$

In the arrangement shown in the figure,if the blocks of masses $m$ and $2m$ are released from rest,find the tension in the string. (Given: $\mu_1 = 2/3$ for mass $m$,$\mu_2 = 1/3$ for mass $2m$,the string is massless and inextensible,and the pulley is frictionless.)

$A$ spring is compressed between two blocks of masses $m_1$ and $m_2$ placed on a horizontal surface as shown in the figure. When the blocks are released,they have initial velocities $v_1$ and $v_2$ as shown. The blocks travel distances $x_1$ and $x_2$ respectively before coming to rest due to friction. The ratio $\left( \frac{x_1}{x_2} \right)$ is