$A$ water pump is driven by a petrol engine that pumps water from a depth of $30 \ m$ at a rate of $0.5 \ m^3/min$. If the efficiency of the pump is $70\%$, what is the power generated by the engine in $W$?

$A$ train is moving along the tracks at a constant speed $u$. $A$ girl on the train throws a ball of mass $m$ straight ahead along the direction of motion of the train with speed $v$ with respect to herself. Then

Two bodies of masses $0.1\, kg$ and $0.4\, kg$ move towards each other with the velocities $1\, m/s$ and $0.1\, m/s$ respectively. After collision they stick together. In $10\, s$,the combined mass travels ............ $m$.

$A$ particle of mass $m$ is projected with a speed $u$ from the ground at an angle $\theta = \frac{\pi}{3}$ w.r.t. horizontal ($x$-axis). When it has reached its maximum height,it collides completely inelastically with another particle of the same mass and velocity $u \hat{i}$. The horizontal distance covered by the combined mass before reaching the ground is

$A$ lead bullet penetrates into a solid object and melts. Assuming that $40 \%$ of its kinetic energy is used to heat it,the initial speed of the bullet is ............ $m \, s^{-1}$.
(Given: Initial temperature of the bullet $= 127^{\circ} C$,
Melting point of the bullet $= 327^{\circ} C$,
Latent heat of fusion of lead $= 2.5 \times 10^{4} \, J \, kg^{-1}$,
Specific heat capacity of lead $= 125 \, J \, kg^{-1} K^{-1}$)

Two objects having masses in the ratio $1:4$ are at rest. When both of them are subjected to the same force separately,they achieve the same kinetic energy during times $t_1$ and $t_2$ respectively. Then the ratio of $t_1$ to $t_2$ is