$A$ spaceship in space sweeps stationary interplanetary dust. As a result,its mass increases at a rate $\frac{dM(t)}{dt} = bv^2(t)$,where $v(t)$ is its instantaneous velocity. The instantaneous acceleration of the spaceship is

$A$ satellite moving with velocity $v$ in a force-free space collects stationary interplanetary dust at a rate of $\frac{dM}{dt} = \alpha v$,where $M$ is the mass (of satellite + dust) at that instant. The instantaneous acceleration of the satellite is:

$A$ $6000 \,kg$ rocket is set for firing. If the exhaust speed is $1000 \,m/s$,how much gas must be ejected each second to supply the thrust needed to overcome the weight of the rocket?

$A$ rocket of mass $1000 \, kg$ exhausts gases at a rate of $4 \, kg/s$ with a velocity of $3000 \, m/s$. The thrust developed on the rocket is ........ $N$.

$A$ rifle of $20 \,kg$ mass can fire $4$ bullets per second. The mass of each bullet is $35 \times 10^{-3} \,kg$ and its final velocity is $400 \,ms^{-1}$. What force must be applied on the rifle so that it does not move backwards while firing the bullets (in $\,N$)?

$A$ rocket has a mass of $100 \, kg$. $90\%$ of this is fuel. It ejects fuel vapours at the rate of $1 \, kg/s$ with a velocity of $500 \, m/s$ relative to the rocket. It is assumed that the rocket is outside the gravitational field. The initial upthrust on the rocket when it just starts moving upwards is ......... $N$.