$A$ ball is thrown from $10 \, m$ height at speed $v_0$ vertically downward. Upon colliding with the surface of the earth,it loses $50 \%$ of its energy and again reaches the same height. The value of $v_0$ is: ................. $m/s$

The friction coefficient between the horizontal surface and each of the blocks shown in the figure is $\mu = 0.2$. The collision between the blocks is perfectly elastic. What is the separation between the blocks when they come to rest? (in $cm$)

Underline the correct alternative:
$(a)$ When a conservative force does positive work on a body,the potential energy of the body increases/decreases/remains unaltered.
$(b)$ Work done by a body against friction always results in a loss of its kinetic/potential energy.
$(c)$ The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system.
$(d)$ In an inelastic collision of two bodies,the quantities which do not change after the collision are the total kinetic energy/total linear momentum/total energy of the system of two bodies.

$A$ bullet of mass $m_1$ is moving with speed $v_0$ and hits a sand bag of mass $m_2$. If the speed of the bullet after passing through the sand bag is $\frac{v_0}{3}$,then the height $h$ up to which the bag rises is (assume,$g=$ acceleration due to gravity).

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 $c_2$?
$(A)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j}$
$(B)$ $\vec{p}_1^{\prime} = c_1 \hat{k}$,$\vec{p}_2^{\prime} = c_2 \hat{k}$
$(C)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j} + c_1 \hat{k}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_2 \hat{j} - c_1 \hat{k}$
$(D)$ $\vec{p}_1^{\prime} = a_1 \hat{i} + b_1 \hat{j}$,$\vec{p}_2^{\prime} = a_2 \hat{i} + b_1 \hat{j}$

$A$ man of mass $60$ $kg$ wants to lose $5$ $kg$ of mass by climbing up and down stairs. Assume that twice the amount of fat is burned while climbing up compared to climbing down. If burning $1$ $kg$ of fat provides $7000$ $kcal$ of energy, how many times must he climb up and down the stairs to lose $5$ $kg$ of mass (in $\times$)? (Assume the energy spent in climbing up is $20$ $kcal$ and climbing down is $10$ $kcal$ per trip).