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 $......^{\circ}$.

An engine pumps up $100 \ kg$ of water through a height of $10 \ m$ in $5 \ s$. Given that the efficiency of the engine is $60\%$. If $g = 10 \ m \ s^{-2}$,the power of the engine is .............. $kW$.

Two bodies $A$ and $B$ of mass $m$ and $2m$ respectively are placed on a smooth floor. They are connected by a spring of negligible mass. $A$ third body $C$ of mass $m$ is placed on the floor. The body $C$ moves with a velocity $v_0$ along the line joining $A$ and $B$ and collides elastically with $A$. At a certain time after the collision,it is found that the instantaneous velocities of $A$ and $B$ are the same and the compression of the spring is $x_0$. The spring constant $k$ will be

$A$ child swings on a swing such that the minimum and maximum heights from the ground are $0.75 \ m$ and $2 \ m$ respectively. The maximum velocity of the swing is ...... $ms^{-1}$.

Two bodies $A$ and $B$ of masses $m$ and $2m$ respectively are placed on a smooth floor. They are connected by a spring. $A$ third body $C$ of mass $m$ moves with velocity $V_0$ along the line joining $A$ and $B$ and collides elastically with $A$ as shown in the figure. At a certain instant of time $t_0$ after the collision,it is found that the instantaneous velocities of $A$ and $B$ are the same. Further,at this instant,the compression of the spring is found to be $x_0$. Determine the spring constant $k$.

From a building, two balls $A$ and $B$ are thrown such that $A$ is thrown upwards and $B$ is thrown downwards (both vertically) with the same speed. If $v_A$ and $v_B$ are their respective velocities on reaching the ground, then: