$A$ 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:

$A$ rocket is fired vertically from the ground with a resultant vertical acceleration of $10 \,m/s^2$. The fuel is exhausted in $1 \,min$ and it continues to move up. What is the maximum height reached (in $\,km$)?

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 $g = 10 \, \text{m s}^{-2}$]

$A$ flexible chain of length $L = 2 \ m$ and mass $M = 1 \ kg$ is initially held in a vertical position such that its lower end just touches a horizontal surface. It is released from rest at time $t = 0$. Assuming that any part of the chain that strikes the plane immediately comes to rest and that the portion of the chain lying on the horizontal surface does not form any heap,find the height of its centre of mass above the surface at the instant $t = 1/\sqrt{5} \ s$ (before it completely comes to rest).

$A$ rocket moves straight upward with zero initial velocity and with an acceleration of $20 \,m/s^2$. It runs out of fuel and stops accelerating at the end of $5^{th} \,sec$. It reaches a maximum height and falls back to the earth. The speed when it hits the ground is (Take $g = 10 \,m/s^2$):

$A$ 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