Q81 / 120 · VITEEE 2017 · Physics
A $5000 \mathrm{~kg}$ rocket is set for vertical firing. The exhaust speed is $800 \mathrm{~m} / \mathrm{s}$. To give an initial upward acceleration of $20 \mathrm{~m} / \mathrm{s}^{2}$, the amount of gas ejected per second to supply the needed thrust will be (Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )
- A$127.5 \mathrm{~kg} / \mathrm{s}$
- B$137.5 \mathrm{~kg} / \mathrm{s}$
- C$155.5 \mathrm{~kg} / \mathrm{s}$
- Correct answerD$187.5 \mathrm{~kg} / \mathrm{s}$
Solution
Given : Mass of rocket $(\mathrm{m})=5000 \mathrm{Kg}$ Exhaust speed $(v)=800 \mathrm{~m} / \mathrm{s}$
Acceleration of rocket $(\mathrm{a})=20 \mathrm{~m} / \mathrm{s}^{2}$
Gravitational acceleration $(\mathrm{g})=10 \mathrm{~m} / \mathrm{s}^{2}$
We know that upward force
$$
\begin{aligned}
\mathrm{F} &=\mathrm{m}(\mathrm{g}+\mathrm{a})=5000(10+20) \\
&=5000 \times 30 &=150000 \mathrm{~N}
\end{aligned}
$$
We also know that amount of gas ejected
$$
\left(\frac{\mathrm{dm}}{\mathrm{dt}}\right)=\frac{\mathrm{F}}{\mathrm{v}}=\frac{150000}{800}=187.5 \mathrm{~kg} / \mathrm{s}
$$
Hence, the correct answer is option: 'D'.