Three blocks $A, B$ and $C$ are kept as shown in the figure. The coefficient of friction between $A$ and $B$ is $0.2$,$B$ and $C$ is $0.1$,and $C$ and the ground is $0.0$. The masses of $A, B$ and $C$ are $3\, kg, 2\, kg$ and $1\, kg$ respectively. $A$ is given a horizontal velocity of $10\, m/s$. Blocks $A, B$ and $C$ always remain in contact and move together as a single system. The total work done by friction will be ........ $J$.

If a rubber ball falls from a height $h$ and rebounds up to a height of $h/2$,the percentage loss of total energy of the initial system and the velocity of the ball just before it strikes the ground,respectively,are:

Two balls,having linear momenta $\vec{p}_1 = p \hat{i}$ and $\vec{p}_2 = -p \hat{i}$,undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1^{\prime}$ and $\vec{p}_2^{\prime}$ be their final momenta. Which of the following option$(s)$ is (are) $NOT ALLOWED$ for any non-zero value of $p, a_1, a_2, b_1, b_2, c_1$ and $c_2$?
$(A)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j}$
$(B)$ $\vec{p}_1^{\prime} = c_1 \hat{k}$,$\vec{p}_2^{\prime} = c_2 \hat{k}$
$(C)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j} - c_1 \hat{k}$
$(D)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_1 \hat{j}$

$A$ smooth sphere of mass $m$ moving with velocity $u$ collides with another smooth sphere of mass $2m$ at rest. What is the range of the velocity $v$ of the second sphere after the collision?

$A$ particle of mass $3 \ kg$ is moved slowly along the path $ABCDE$ from $A$ to $E$. The heights of $B$,$C$,and $D$ are $5 \ m$,$4 \ m$,and $6 \ m$ respectively. The total path length is $20 \ m$,the horizontal distance $AE = 10 \ m$,and the coefficient of friction $\mu = 0.6$. Calculate the work done in $J$ to slowly move the mass from $A$ to $E$. (Take $g = 10 \ m/s^2$)

Two bodies have kinetic energies in the ratio $16: 9$. If they have the same linear momentum,the ratio of their masses respectively is