Question

A $ 5000 \,kg $ rocket is set for vertical firing. The exhaust speed is $ 800 \,ms^{−1} $ . To give an initial upward acceleration of $ 20 \,ms^{−2} $ , the amount of gas ejected per second to supply the needed thrust will be $ (g = 10\,ms^{−2}) $

• A. $ 127.5\,kg\, s^{−1} $
• B. $ 187.5\,kg\, s^{−1} $
• C. $ 185.5\,kg\, s^{−1} $
• D. $ 137.5\,kg\, s^{−1} $

Answer 1

Answer: (B)

Thrust force on the rocket
$F_{t}=v_{r}\left(-\frac{d m}{d t}\right)\,\,\,\, $(upwards)
Weight of the rocket
$w=m g\,\,\,\, $(downwards)
Net force on the rocket
$F_{\text {net }} =F_{t}-w $
$\Rightarrow m a =v_{r}\left(\frac{-d m}{d t}\right)-m g $
$\Rightarrow \left(\frac{-d m}{d t}\right) =\frac{m(g+a)}{v_{r}}$
$\therefore $ Rate of gas ejected per second
$=\frac{5000(10+20)}{800}=\frac{5000 \times 30}{800} $
$=187.5\, kg \,s ^{-1}$