The initial mass of a rocket is $1000 \, \text{kg}$. Calculate at what rate the fuel (in $\text{kg s}^{-1}$) should be burnt so that the rocket is given an acceleration of $20 \, \text{m s}^{-2}$. The gases come out at a relative speed of $500 \, \text{m s}^{-1}$ with respect to the rocket. [Use $g = 10 \, \text{m s}^{-2}$]

$A$ satellite in force-free space sweeps stationary interplanetary dust at a rate of $\frac{d M}{d t} = \alpha v$,where $M$ is the mass,$v$ is the speed of the satellite,and $\alpha$ is a constant. The acceleration of the satellite is:

$A$ metal block of $2 \,kg$ is at rest on a smooth plane. It is struck by a water jet releasing water at a rate of $1 \,kg \,s^{-1}$ with a speed of $5 \,ms^{-1}$. The acceleration of the block is: (in $\,ms^{-2}$)

$A$ soldier with a machine gun,falling from an airplane,gets detached from his parachute. He is able to resist the downward acceleration if he shoots $40$ bullets per second at a speed of $500 \,m/s$. If the mass of a bullet is $49 \,g$,what is the mass of the man with the gun in $kg$? Ignore air resistance and assume the acceleration due to gravity $g = 9.8 \,m/s^2$.

$A$ sand dropper drops sand of mass $m(t)$ on a conveyer belt at a rate proportional to the square root of speed $v$ of the belt,i.e.,$\frac{dm}{dt} \propto \sqrt{v}$. If $P$ is the power delivered to run the belt at constant speed,then which of the following relationships is true?

In a rocket,fuel burns at the rate of $1\; kg/s$. This fuel is ejected from the rocket with a velocity of $60\; km/s$. This exerts a force on the rocket equal to ......... $N$.