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A rocket consumes 20 kg fuel per second. The exhaust gases escape at a speed at $1000\;m{s^{ - 1}}$ relative to the rocket. Calculate the velocity acquired by the rocket, when its mass reduces to $1/100$ of its initial mass.
A) 4.5 km/sec
B) 4.6 km/sec
C) 4.06 km/sec
D) 4.05 km/sec

Last updated date: 13th Jun 2024
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Hint: The rocket propels upwards due to the thrust produced by the burning of the fuel. This thrust onto the ground causes a force upwards which propels the rocket upwards. Due to the force acting upwards the rocket will be accelerated upwards. This up thrust is produced by the burning of fuel in the rocket.

Complete step by step answer:
The velocity V acquired when the final mass is $1/100$ of the initial mass is :
$V = u\ln \left( {\dfrac{M}{m}} \right)$
Here u is the initial velocity, m is the final mass and M is the initial mass.
V = 1000\;\ln \left( {100} \right)\\
   = 4606\;m/s\\
   = 4.6\;km/s

Therefore, the correct option is (B).

Additional Information: To understand the science behind the Rocket, you can take an example of a gunshot. Shooting a gun also demonstrates the application of conservation of momentum. As we pull the trigger, the bullet comes out at a very high speed, but we also observe a recoil of the gun. This happens to conserve momentum. The momentum gained by the bullet is equal to and also the reason for the recoil of the gun. Same as this, the gases inside a rocket are made to propel out of the Rocket at a very high speed. This, in turn, gives a push to the rocket in the opposite direction to conserve the momentum. Thus a lot of fuel needs to be burned to provide the rocket a sufficient amount of force to escape the earth's atmosphere.

Note: The force that propels the rocket upwards is the consequence of Newton’s Third Law. The force of thrust on to the ground and the force on the rocket is a Newtonian pair of forces which are equal and opposite. The force acting as thrust is due to the mass dissipation of the fuel per time which with the velocity of the exhaust produces a thrust force is responsible for the propulsion.