$A$ block of mass $m = 5 \text{ kg}$ is released from the top of an inclined plane as shown in the figure. The inclined plane has a length of $10 \text{ m}$ and an angle of $30^{\circ}$. The horizontal surface has a coefficient of kinetic friction $\mu = 0.5$ and a length of $2 \text{ m}$ before the spring of spring constant $k = 100 \text{ N/m}$. Calculate the maximum compression $x$ in the spring.

$A$ ball of mass $2m$ is moving with velocity $v$ on a smooth surface and collides elastically head-on with another ball of mass $m$ which is at rest. If the ball of mass $m$ reaches up to the top of a frictionless elevated plane of height $h$,then the velocity $v$ of the heavy ball must be

$A$ sphere of mass $m$,moving with velocity $V$,enters a hanging bag of sand and stops. If the mass of the bag is $M$ and it is raised by height $h$,then the velocity of the sphere was

$A$ ball loses $15.0\%$ of its kinetic energy when it bounces back from a concrete wall. With what speed must you throw it vertically down from a height of $12.4\, m$ to have it bounce back to the same height (ignore air resistance)? ............. $m / s$

Two bodies of masses $m$ and $2m$ are attached to the two ends of an ideal spring. The spring is compressed. The total energy stored in the spring is $60 \ J$. If the spring is released,then:

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