- #1
DanielA
- 27
- 2
Homework Statement
A rocket (initial mass ##m_o##, constant exhaust velocity ##v_{ex}## needs to use its engines to hover stationary, just above the ground.
a) If it can afford to burn no more than a mass ##\lambda m_o## of its fuel (##\lambda \lt##), for how long can it hover?
b) If ##v_{ex} = 3000 m/s ~\text{and} \lambda \approx 0.10## for how long could the rocket hover just above the Earth's surface?
Homework Equations
The question didn't give any explicit equations, but the question should use $$\dot p = m dv + dm v_{ex} = F^{ext} dt $$ where ##dm v_{ext} = thrust##
which I believe is the rocket equation for a rocket moving vertically. I got it from my book, though they continued the example with no gravity or air ressitance so ##F^{ext} = 0##
The Attempt at a Solution
To begin, I believe I have a finished solution, but since I'm used to having a professor around to check my work, I'm not confident I'm doing it right as it seems too simple.
a)
Since there is no motion (and we haven't learned air resistance for a rocket) the only force ##F^{ext}## is gravity and ##F^{ext} = mg## where m is the mass of the rocket.
So, we want to solve for t. Let's use my equation above and integrate it .
$$
mgdt = mdv + dm v_{ex}
\\ \int_{v_0}^v dv = \int_0^t g\, dt - v_{ex} \int_{m_0}^m {\frac {dm} m}
\\ v-v_0 = gt - v_{ex} \ln {\frac m {m_0}}
\\ v = 0, v_0 = 0, m = \lambda m_0
\\ \text{since the rocket is motionless throughout all of this and the maximum mass loss is given in the question}
\\ t = \frac {v_{ex}} g \ln (\lambda)
$$
b)
This part is just plug and chug using the above equation. My answer is 704.15 seconds (g = -9.81 m/s), which seems excessively long to me, but I don't have any frame of reference on what is realistic for this.