Consider a rocket is being fired. The kinetic energy of the rocket is increased by $16$ times, whereas its total mass is reduced by half through the burning of fuel. The factor by which its momentum increases is

$A$ flat plate moves normally with a speed $v_1$ towards a horizontal jet of water of uniform area of cross-section. The jet discharges water at the rate of volume $V$ per second at a speed of $v_2$. The density of water is $\rho$. Assume that water splashes along the surface of the plate at right angles to the original motion. The magnitude of the force acting on the plate due to the jet of water is

$A$ 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 $\,N$)?

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

$A$ rocket which has a mass of $3.5 \times 10^4 \,kg$ is blasted upwards with an initial acceleration of $10 \,m/s^2$. The initial thrust of the blast is-

$A$ balloon with mass $m$ is descending vertically with an acceleration $a$ (where $a < g$). The mass to be removed from the balloon so that it starts moving vertically up with an acceleration $a$ is: