Question

For a rocket propulsion velocity of exhaust gases relative to the rocket is $2\text{ km/s}$. If the mass of the rocket is 1000 kg. Then the rate of fuel consumption for a rocket to rise up with an acceleration of $4.9\text{ m/}{{\text{s}}^{2}}$ will be,
A.) $12.25\text{ kg/s}$
B.) $17.5\text{ kg/s}$
C.) $7.35\text{ kg/s}$
D.) $5.2\text{ kg/s}$

Answer 1

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 upthrust is produced by the burning of fuel in the rocket.

Complete step by step answer:
In the problem, the mass of the rocket is 1000 kg, the acceleration it achieves is $4.9\text{ m/}{{\text{s}}^{2}}$. So, the relative velocity of the exhaust gas is $2\text{ km/s}$. So, the rate of fuel consumption is given by newton’s second law,

We know that according to Newton’s second law, the force acting on a body is the rate of change of momentum. So, we can write,

$F=\dfrac{dP}{dt}$

The momentum of a body is expressed as the product of the mass of the body and the velocity of the body. $\left( P=mv \right)$. So, substituting the momentum in the above equation we get,

$F=\dfrac{d\left( mv \right)}{dt}=m\dfrac{dv}{dt}+v\dfrac{dm}{dt}$

Here the exhaust velocity remains constant, so the term $\dfrac{dv}{dt}$ is zero.

$\therefore F={{v}_{ex}}\dfrac{dm}{dt}$

This force is providing the force that accelerates the rocket upwards. There is also the force of gravity acting downwards against the motion of the rocket. So, the sum of theses force should be equal to the force on the rocket.

${{v}_{ex}}\dfrac{dm}{dt}+\left( -{{F}_{g}} \right)={{F}_{r}}$
${{v}_{ex}}\dfrac{dm}{dt}-Mg=M{{a}_{r}}$

Where,

${{v}_{ex}}$ is the exhaust velocity.

M is the mass of the rocket.
${{a}_{r}}$ is the acceleration of the rocket.
g is the acceleration due to gravity.

Substituting the values of mass, velocity and acceleration in the above equation, we get,

$\left( 2000m/s \right)\dfrac{dm}{dt}-\left( 1000kg\times 9.8m/{{s}^{2}} \right)=\left( 1000kg \right)\times \left( 4.9m/{{s}^{2}} \right)$
$\dfrac{dm}{dt}=\dfrac{3kg\times 4.9m/{{s}^{2}}}{2m/s}$
$\dfrac{dm}{dt}=7.35\text{ }kg/s$

So, the answer to the question is option (C).

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.