$A$ block of mass $M = 10\,kg$ rests on a horizontal table. The coefficient of friction between the block and the table is $\mu = 0.05.$ When hit by a bullet of mass $m = 50\,g$ moving with speed $v,$ which gets embedded in it,the block moves and comes to a stop after moving a distance of $2\,m$ on the table. If a freely falling object were to acquire speed $\frac{v}{10}$ after being dropped from height $H,$ then neglecting energy losses and taking $g = 10\,m/s^2,$ the value of $H$ is close to ................. $km$.

$A$ point particle of mass $m$ moves along the uniformly rough track $PQR$ as shown in the figure. The coefficient of friction between the particle and the rough track equals $\mu$. The particle is released from rest from the point $P$ and it comes to rest at a point $R$. The energies lost by the particle over the parts $PQ$ and $QR$ of the track are equal to each other,and no energy is lost when the particle changes direction from $PQ$ to $QR$. The values of the coefficient of friction $\mu$ and the distance $x (= QR)$ are,respectively,close to

$A$ $100 \, kg$ 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 $, m/s$)

$A$ mass $2 \sqrt{3} \,kg$ is acted upon by two forces which are inclined to each other at $60^{\circ}$ and each of magnitude $1 \,N$. The acceleration of that mass in $SI$ system is $\left[\sin 30^{\circ}=\cos 60^{\circ}=0.5\right]$ (in $\,m / s^{2}$)

Three blocks $A, B$ and $C,$ of masses $4 \, kg, 2 \, kg$ and $1 \, kg$ respectively,are in contact on a frictionless surface,as shown. If a force of $14 \, N$ is applied on the $4 \, kg$ block,then the contact force between $A$ and $B$ is .......... $N$.

In the figure,a ladder of mass $m$ is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is $\mu_1$ and that between the floor and the ladder is $\mu_2$. The normal reaction of the wall on the ladder is $N_1$ and that of the floor is $N_2$. If the ladder is about to slip,then