Question

A rocket has a mass of $1000\text{kg}$ with $50,000\text{kg}$ of fuel being stored on this rocket. The nozzle of the rocket lets out the exhaust gases at a speed of $500\text{m/s}$. If the rocket consumes fuel at the rate of $100\text{kg/s}$, what is the acceleration of the rocket after $5\text{minutes}$ in a gravity free space?

• A. $10{\text{m/s}}^2$
• B. $3.95{\text{m/s}}^2$
• C. $2.38{\text{m/s}}^2$
• D. $1.56{\text{m/s}}^2$

Answer 1

The correct option is C $2.38{\text{m/s}}^2$

Mass of the rocket $=1000\text{kg}$
Mass of fuel stored $=50000\text{kg}$
Total mass of rocket + fuel $m=(50000+1000)\text{kg}=51000\text{kg}$
Consumption rate of fuel $=\frac{dM}{dt}=100\text{kg/s}$
Exhaust velocity of gases, $u=500\text{m/s}$
Time, $T=5\text{minutes}=300\text{seconds}$
From Newton's second law, we have,
$F=ma\Rightarrow a=\frac{F}{m}$
Here $F$ is the force exerted on the rocket.
$a=\frac{F}{m}=\frac{u\frac{dM}{dt}}{m-\frac{dM}{dt}\times T}=\frac{500\times 100}{51000-100\times 300}$
$\Rightarrow a=\frac{50000}{51000-30000}=2.38{\text{m/s}}^2$
Hence acceleration of the rocket after 5 minutes is $2.38{\text{m/s}}^2$
Hence the correct option is (c).