Question

A rocket of mass $3500\text{kg}$ has to achieve a velocity of $1000\text{m/s}$ over a launch time of $10$ minutes. The nozzle of the rocket is designed to let out exhaust gases at a velocity of $300\text{m/s}$ w.r.t the rocket. What is the amount of fuel that is consumed during the launch?

• A. $81660\text{kg}$
• B. $66666.66\text{kg}$
• C. $90000\text{kg}$
• D. $88666.66\text{kg}$

Answer 1

The correct option is A $81660\text{kg}$

Initial mass of the rocket $=m=3500\text{kg}$
Acceleration of the rocket over the lauch time
$=a=\frac{v_f}{t}=\frac{1000\text{m/s}}{10\times 60\text{s}}=1.666{\text{m/s}}^2$
Relative velocity of exhaust gases $v_r=300\text{m/s}$

We have the equation,
$v_r\frac{dm}{dt}-mg=ma$
$\Rightarrow 300\times \frac{dm}{dt}-3500\times 10=3500\times 1.666$
$\Rightarrow \frac{dm}{dt}=\frac{3500\times 11.666}{300}=136.10\text{kg/s}$
Amount of fuel consumed over $10$ minutes
$M=136.10\times 10\times 60=81660\text{kg}$
Hence $81660\text{kg}$ is consumed over ten minutes.