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

Hint: 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
\[\begin{align}
  & {{f}_{net}}={{f}_{t}}-w \\
 & \Rightarrow ma={{v}_{r}}\left( \dfrac{-dm}{dt} \right)-mg \\
 & \Rightarrow \dfrac{-dm}{dt}=\dfrac{m\left( g+a \right)}{{{v}_{r}}} \\
\end{align}\]
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.