Question

A 5000 kg rocket is set for vertical firing.
The exhaust speed is $800m/s$. To give an upward acceleration of $20m/s$, the amount of gas ejected per second to supply the needed thrust is $\left( g=10m/{{s}^{2}} \right)$.
A. $127.5kg{{s}^{-1}}$
B. $137.5kg{{s}^{-1}}$
C. $187.5kg{{s}^{-1}}$
D. $188.5kg{{s}^{-1}}$

Answer 1

Since, a rocket of mass 500 kg is set for a vertical firing and its exhaust speed is 800 m/s. upward acceleration is 20m/s. Then we use the formula of thrust on the rocket to find the amount of gas ejected per second to supply the needed thrust.

Complete answer:
Given that
Mass of the rocket m = 500kg
Acceleration of the rocket $a=20m/{{s}^{2}}$
Speed of the rocket $v=800m/s$
Also, $g=10m/{{s}^{2}}$
As we know that the force on the rocket is given by
${{f}_{z}}={{v}_{r}}\left( \dfrac{-dm}{dt} \right)$
So, Net force on the rocket

& {{f}_{net}}={{f}_{t}}-w \\\ & \Rightarrow ma={{v}_{r}}\left( \dfrac{-dm}{dt} \right)-mg \\\ & \Rightarrow \dfrac{-dm}{dt}=\dfrac{m\left( g+a \right)}{{{v}_{r}}} \\\ \end{aligned}$$ Rate of gas ejected per second$$\dfrac{-dm}{dt}=\dfrac{5000\left( 10+20 \right)}{800}$$ $$\Rightarrow \dfrac{-dm}{dt}=187.5kg/s$$ Therefore, the amount of Second is 187.5 kg /s. **So, the correct answer is “Option C”.** **Note:** As we know that the Thrust force on the rocket is given by$${{f}_{t}}={{v}_{r}}\left( \dfrac{-dm}{dt} \right)$$ and Net force on the rocket $${{f}_{net}}=ma$$. So, use the formula to calculate the amount of gas ejected per second i.e. rate is given by $$\dfrac{-dm}{dt}$$ Be careful during the calculation and put the value at its exact time.