Question

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

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

Answer 1

The correct option is B

$187.5kg{s}^{-1}$

Step 1: Given data

Mass of the rocket, $m=5000kg$

The exhaust speed,$v=800m{s}^{-1}$

Initial upward acceleration of rocket, $a=20m{s}^{-2}$

Step 2: To find the force due to gravitation

Force due to gravitation, $F=mg$

$$\begin{gathered} =5000kg\times 10m{s}^{-2} \\ =50000N \end{gathered}$$

Step 3: To find the net force which responsible for acceleration

Net force responsible for acceleration, $F_{net}=5000kg\times 20m{s}^{-2}$

$=100000N$

Step 3: To find the amount of gas ejected per second

Value of thrust force, $TF=v\frac{dm}{dt}$

Net force responsible for acceleration, $F_{Net}=TF-F$

$$\begin{gathered} 100000=800m{s}^{-1}\frac{dm}{dt}-50000N \\ 100000=800\frac{dm}{dt}-50000 \\ 150000=800\frac{dm}{dt} \\ \frac{dm}{dt}=187.5kg{s}^{-1} \end{gathered}$$

Hence, option C is correct.