Question

A rocket of mass $120\text{kg}$ has $280\text{kg}$ of fuel is ready for the take off. The relative exhaust velocity of the fuel is $1.8\text{km/s}$ and the rate of consumption of the fuel is $2\text{kg/s}$. Calculate the ultimate vertical speed gained by the rocket. (Take $g=9.8{\text{m/s}}^2$)

• A. $3.54\text{km/s}$
• B. $1.79\text{km/s}$
• C. $0.795\text{km/s}$
• D. $0\text{km/s}$

Answer 1

The correct option is C $0.795\text{km/s}$

Given,
Mass of the rocket without fuel $=120\text{kg}$
Mass of the fuel in the rocket $=280\text{kg}$
$\therefore$ Total initial mass of the rocket $m_0=400\text{kg}$
Exhaust velocity of the fuel $\left(v_r\right)=1.8\text{km/s}$
Rate of consumption of fuel $\mu =2\text{kg/s}$
Then, time at which the fuel will be finished $\left(t\right)=\frac{280}{2}=140\text{s}$

Final velocity of the rocket at $t=140\text{s}$
$v=u-gt+v_r\ln \left(\frac{m_0}{m}\right)$
$=0-9.8\times 140+1.8\times {10}^3\ln \left(\frac{400}{120}\right)$
$=-1372+2167$
$=795\text{m/s}=0.795\text{km/s}$