Question

A rocket motor consumes $100\, kg$ of fuel per second exhausting it with a speed of $5 \,km / s$. The speed of the rocket when its mass is reduced to $\frac{1^{\text {th }}}{20}$ of its initial mass, is (Assume initial speed to be zero and ignored gravitational and viscous forces.)

• A. $20 \,km / s$
• B. $40 \ln (2) \,km / s$
• C. $5\, \ln (20) \,km / s$
• D. $10 \ln (10) \,km / s$

Answer 1

Answer: (C)

$5\, \ln (20) \,km / s$

Answer 2

Velocity of a rocket at any time $t$,
$v=u\left(\frac{m_{0}}{m}\right)-gt$
where, $u=$ speed of exhausted gases,
$m_{0}=$ initial mass of the rocket
and $m=$ mass of the rocket at time $t$
Given, fuel burned rate, $\left(\frac{d m}{d t}\right)=100 \,kg / s , u=5\,km / s$
Then, $v=5 \ln \left(\frac{m_{0}}{m}\right)-g t$
As, it is given that the gravitational force is ignored and mass of rocket is reduced to $\frac{1}{20}$ th of it's initial mass i.e, $m=\frac{1}{20} m_{0}$.
or $\frac{m_{0}}{m}=20$
So, $v=5 \ln (20)\, km / s$