Question

The exhaust velocity of gases with respect to a small rocket of mass 25 kg is $28\times 10^2 m/s$. At what rate the fuel must burn so that it may rise up with an acceleration of $9.8 m/s^2$ ?
A. 175 kg/s
B. 1.75 kg/s
C. 0.175 kg/s
D. Zero

Answer 1

Hint: A rocket is an aircraft/ spacecraft or a vehicle which is basically used to launch anything from the earth to space. The material to be launched is known as ‘payload’. We can explore different parts of the universe using rockets.

Formula used: $F_T = v_r\dfrac{dm}{dt}$

Complete step by step answer:
The launch of a rocket is based on Newton’s laws of motion. The main fact behind it is that the fuel is ejected from the bottom part of the rocket at a very high rate so that the rocket experiences an upward thrust (due to momentum of fuel) and this thrust when gets greater than the weight of the rocket, the rocket gets launched.
The thrust force which the fuel exerts on the rocket is given by $F_T = v_r\dfrac{dm}{dt}$, where $v_r$ is the velocity of fuel with respect to rocket and $\dfrac{dm}{dt}$ is the rate of burn of fuel.
Given $v_r = 28 \times 10^2 m/s = 2800 m/s$
Now, force that it has to experience = $F_{net} = m\times a = 25 \times 9.8 = 245 N$
But, gravity will also play its role here, hence;
$F_T - mg = F_{net}$
$F_T = F_{net} + mg$
$F_{T} = 245 + 25 \times 9.8 = 245 + 245 = 490 N$
Now, putting the values in the equation: $F_T = v_r\dfrac{dm}{dt}$, we get;
$490 = \dfrac{dm}{dt} \times 2800$
$\implies \dfrac{dm}{dt} = 0.175 kg/s$

So, the correct answer is “Option C”.

Note: The phenomenon of rocket launching is based on Newton’s third law which states that “Every action has an equal and opposite reaction. The fuel is ejected with respect to the rocket continuously at a very high speed and hence the fuel in turn applies force upwards on the rocket, which lifts it. Hence in this way, the rocket gets launched. Chance of mistake is that students might think that the rocket being heavy, its weight could be neglected. But it is not so. Unless specified, we have to always consider the effect of gravity.