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Piamedes
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Homework Statement
A rocket (initial mass m0) needs to use its engines ot hover stationary, just above the ground. If it can afford to burn no more than a mass (lamda)m0 of its fuel, for how long can it hover? If the exhaust velocity is 3000 m/s and lambda is 10% how long can the rocket hover?
Homework Equations
Thrust = [tex] v_{ex} \frac{dm}{dt} [/tex]
[tex] v_{ex} [/tex] is the exhaust velocity
The Attempt at a Solution
So starting off with the basic sum of forces equals zero:
[tex] \dot{m}v_{ex} = mg [/tex]
[tex] \frac{dm}{dt} = \frac{mg}{v_{ex}} [/tex]
[tex] \frac{dm}{m} = \frac{g dt}{v_{ex}} [/tex]
[tex] \frac{g}{v_{ex}} \int_{0}^{t} dt = \int_{m_{0}}^{m_{0} - \lambda m_{0}} \frac{dm}{m}[/tex]
[tex] \frac{g}{v_{ex}} t = ln(m) ]_{m_{0}}^{m_{0} - \lambda m_{0}} [/tex]
[tex] \frac{g}{v_{ex}} t = ln [m_{0} - \lambda m_{0}] - ln [m_{0}] [/tex]
[tex] \frac{g}{v_{ex}} t = ln [\frac{m_{0} - \lambda m_{0}}{m_{0}}] [/tex]
[tex] \frac{g}{v_{ex}} t = ln [ 1 - \lambda] [/tex]
[tex] t = \frac{v_{ex}}{g} ln [ 1 - \lambda] [/tex]
And here is where I encounter an issue. Since lambda will always be positive, regardless of how much fuel the rocket can expend this equation will give a negative time. Does anyone see where I went wrong with the calculations, or are my original assumptions wrong?
Thanks for any and all help