Question

A rocket is in gravity free space. A scientist wants the acceleration of the rocket to be $2{\text{m/s}}^2$. Initial mass of the rocket is $1000\text{kg}$ and the speed of ejection of burnt fuel w.r.t the rocket is $500\text{m/s}$. Find the rate of consumption of fuel $\left(\text{in kg/s}\right)$, so that the rocket will have an acceleration of $2{\text{m/s}}^2.$

• A. $2\text{kg/s}$
• B. $4\text{kg/s}$
• C. $3\text{kg/s}$
• D. $1\text{kg/s}$

Answer 1

The correct option is B $4\text{kg/s}$

Given, initial mass of the rocket $m=1000\text{kg}$
Acceleration of the rocket $a=2{\text{m/s}}^2$
Relative speed of ejection of burnt fuel $v_r=500\text{m/s}$


From equation of variable mass system,
$F_t+F_{ext}=ma$
It is gravity free space, hence $F_{ext}=-mg=0$
& $F_t=-v_r\frac{dm}{dt}$

i.e $v_r(-\frac{dm}{dt})=ma$
$500\times (-\frac{dm}{dt})=1000\times 2$
(relative velocity of ejection downward)
$\Rightarrow \frac{dm}{dt}=-4\text{kg/s}$
(-ve sign signifies mass is decreasing)