$A$ rocket has an initial mass of $20 \times 10^3 \, kg$. If it is to blast off with an initial acceleration of $4 \, m/s^2$,the initial thrust needed is $(g = 10 \, m/s^2)$.

$A$ rocket accelerates straight up by ejecting gas downwards. In a small time interval $\Delta t$,it ejects a gas of mass $\Delta m$ at a relative speed $u$. Calculate the kinetic energy $(KE)$ of the entire system at time $t + \Delta t$ and $t$,and show that the device that ejects gas does work equal to $(\frac{1}{2}) \Delta m u^2$ in this time interval (ignoring gravity).

Sand is deposited at a uniform rate of $20 \ kg/s$ and with negligible kinetic energy on an empty conveyor belt moving horizontally at a constant speed of $10 \ m/min$. Find the force required to maintain the constant velocity of the conveyor belt.

$A$ satellite in force-free space sweeps stationary interplanetary dust at a rate $\frac{dM}{dt} = \alpha v$,where $M$ is the mass,$v$ is the velocity of the satellite,and $\alpha$ is a constant. What is the deceleration of the satellite?

$A$ balloon and its content having mass $M$ is moving up with an acceleration $a$. The mass that must be released from the content so that the balloon starts moving up with an acceleration $3a$ will be: (Take $g$ as acceleration due to gravity)

$A$ metal block of mass $4 \ kg$ is at rest on a frictionless surface. It is struck by a jet of water releasing $2 \ kg \ s^{-1}$ at a speed of $10 \ m \ s^{-1}$. The acceleration of the block is (in $m \ s^{-2}$)