The initial mass of a rocket is 1000 kg. Calculate at what rate the fuel should be burnt, so that the rocket is given an acceleration of 20 ms⁻². The gases come out at a relative speed of 500 ms⁻¹ with respect to the rocket [Use, g = 10 m/s² ]

Given, M = 1000 kg, a = 20 m/s²

v_relative = 500 m/ s, g = 10 m/s²

The given situation is shown below

$\begin{array}{l} {\mathrm{F}}_{\text{thrust }}=\frac{\mathrm{d}}{\mathrm{dt}}\left({\mathrm{Mv}}_{\text{relative }}\right)\\ \Rightarrow {\mathrm{F}}_{\text{thrust }}={\mathrm{v}}_{\text{relative }}\left(\frac{\mathrm{dM}}{\mathrm{dt}}\right)\end{array}$

By Newton’s second law of motion,

$\begin{array}{l}\Rightarrow {\mathrm{F}}_{\text{thrust }}-\mathrm{Mg}=\mathrm{Ma}\\ \Rightarrow {\mathrm{v}}_{\text{relative }}\left(\frac{\mathrm{dM}}{\mathrm{dt}}\right)-\mathrm{Mg}=\mathrm{Ma}\end{array}$

$\begin{array}{l}\Rightarrow 500\left(\frac{\mathrm{dM}}{\mathrm{dt}}\right)-1000\times 10=1000\times 20\\ \Rightarrow 500\left(\frac{\mathrm{dM}}{\mathrm{dt}}\right)=1000(20+10)\\ \Rightarrow 500\left(\frac{\mathrm{dM}}{\mathrm{dt}}\right)=1000\times 30\\ \Rightarrow \frac{\mathrm{dM}}{\mathrm{dt}}=\frac{1000\times 30}{500}=60\\ \Rightarrow \frac{\mathrm{dM}}{\mathrm{dt}}=60\mathrm{kg}/\mathrm{s}\end{array}$