$A$ trolley containing water has a total mass of $4000\, kg$ and is moving at a speed of $40\, m/s$. Water starts leaking from a hole at the bottom of the trolley at a rate of $8\, kg/s$. What is the speed of the trolley after $50\, s$?

$A$ rocket lifts off from the ground and accelerates upwards at $1 \ m \ s^{-2}$. $20 \ s$ after liftoff, a piece breaks off from the bottom of the rocket. After breaking off, how much time does it take approximately to reach the ground (in $s$)? (Take $g = 10 \ m \ s^{-2}$)

$A$ railway wagon of mass $5 \times 10^3 \ kg$ is moving with an initial velocity of $1.2 \ m/s$ on frictionless rails. Rain falls vertically into the wagon. When $10^3 \ kg$ of water has collected in the wagon,what is the change in the kinetic energy of the wagon in $J$?

$A$ machine gun is mounted on a $2000 \,kg$ car on a horizontal frictionless surface. At some instant,the gun fires bullets of mass $10 \,g$ with a velocity of $500 \,m/s$ with respect to the car. If the number of bullets fired per second is $10$,the acceleration of the car will be ..... $m/s^2$.

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

$A$ heap of rope with mass density $\lambda$ (per unit length) lies on a table. You grab one end and pull horizontally with constant speed $v$,as shown in the figure. (Assume that the rope has no friction with itself in the heap.) The force that you must apply to maintain the constant speed $v$ is