$A$ rocket with an initial mass $m_0$ is going up with a constant acceleration $a$ by exhausting gases with a velocity $v$ relative to the rocket motion. The mass of the rocket at any instant of time $t$ is (assume that no other forces act on it).

If the force on a rocket having an exhaust velocity of $300 \, m/s$ is $210 \, N$,then the rate of combustion of the fuel is ........... $kg/s$.

$A$ 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 being dropped from a stationary dropper at a rate of $0.5 \, kg s^{-1}$ on a conveyor belt moving with a velocity of $5 \, m s^{-1}$. The power needed to keep the belt moving with the same velocity will be: (in $, W$)

$A$ balloon has a mass of $10\,g$ in air. The air escapes from the balloon at a uniform rate with a velocity of $4.5\,cm/s$. If the balloon shrinks completely in $5\,s$,then the average force acting on the balloon will be (in $dyne$).

$A$ rocket is fired upward from the earth's surface such that it creates an acceleration of $19.6 \,m/s^2$. If after $5 \,s$ its engine is switched off,the maximum height of the rocket from the earth's surface would be.........$m$