$A$ 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 ......... $N$.

$A$ stationary rocket of initial mass $m_0$ is expelling exhaust gas at a steady speed $v_e$ with respect to itself. There is no external force on the rocket. When its mass is reduced to $\frac{m_0}{2}$,the velocity attained by the rocket will be:

$A$ 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).

In a rocket, fuel burns at the rate of $1 \,kg / s$. This fuel is ejected from the rocket with a velocity of $60 \,km / s$. The force exerted on the rocket by this is (in $\,N$)

$A$ rocket motor consumes $100 \ kg$ of fuel per second,exhausting it with a speed of $5 \ km/s$. The speed of the rocket when its mass is reduced to $\frac{1}{20}$ of its initial mass is (Assume initial speed to be zero and ignore gravitational and viscous forces.)

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 $, W$)