$A$ circular well of diameter $2 \ m$ has water up to the ground level. If the bottom of the well is at a depth of $14 \ m$,the time taken in seconds to empty the well using a $1.4 \ kW$ motor is (Acceleration due to gravity $= 10 \ m \ s^{-2}$)

Particle $A$ of mass $m_{1}$ moving with velocity $(\sqrt{3} \hat{i} + \hat{j}) \, m/s$ collides with another particle $B$ of mass $m_{2}$ which is at rest initially. Let $\vec{V}_{1}$ and $\vec{V}_{2}$ be the velocities of particles $A$ and $B$ after collision,respectively. If $m_{1} = 2m_{2}$ and after collision $\vec{V}_{1} = (\hat{i} + \sqrt{3} \hat{j}) \, m/s$,the angle between $\vec{V}_{1}$ and $\vec{V}_{2}$ is $......^{\circ}$.

$A$ body of mass $5\,kg$ is moving with a momentum of $10\,kg\,m/s$. Now a force of $2\,N$ acts on the body in the direction of its motion for $5\,s$. The increase in the kinetic energy of the body is $...........J$.

Two massless strings of length $5\, m$ hang from the ceiling very near to each other as shown in the figure. Two balls $A$ and $B$ of masses $0.25\, kg$ and $0.5\, kg$ are attached to the strings. The ball $A$ is released from rest at a height of $0.45\, m$ as shown in the figure. The collision between the two balls is completely elastic. Immediately after the collision,the kinetic energy of ball $B$ is $1\, J$. The velocity of ball $A$ just after the collision is

$A$ mass of $1 \; kg$ collides elastically with a stationary mass of $5 \; kg$. After the collision,the $1 \; kg$ mass moves in the opposite direction with a velocity of $2 \; ms^{-1}$. Which of the following statements is correct for this system of two masses?

$A$ 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: ................. $m/s$