DGL aufstellen < gewöhnliche < Differentialgl. < Analysis < Hochschule < Mathe < Vorhilfe
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| Aufgabe 1 | B-Exercises
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2.
A rocket has a mass M, which includes a mass m of a fuel mixture. During the burning process the combustion products are discharged at a velocity q > 0 relative to the rocket. This burning involves a loss per second of a mass p of the fuel mixture. Neglecting all external forces except a constant gravitational, show that the maximum theoretical height attained by the rocket is
$ [mm] \bruch{qm}{p}+\bruch{qM}{p}\cdot{}ln\left( \bruch{M-m}{M}\right)+\bruch{q^2}{2g}\cdot{}ln^2\left( \bruch{M-m}{M}\right) [/mm] $ |
| Aufgabe 2 | 3. In addition to the gravitational force acting on the rocket of Exercise 2, there is a force due to air resistance which is proportional to the instantaneous velocity of the rocket.
(a) Find the velocity of the rocket at any time assuming that its initial velocity is zero.
(b) Determine the height of the rocket at any time.
(c) Find the maximum theoretical height attained. |
Hallo,
es geht um Aufgabe 3. Ich wollte fragen, ob ich die DGL richtig aufgestellt habe.
M : gesamte Raketenmasse
m: Treibstoffmasse
q : const. Geschwindigkeit der Verbrennungsgase relativ zur Rakete in [m/s]
p : Massenstrom in [kg/s]
g = const.
$ [mm] F=m\cdot{}a=F_{Schub}-G-k\cdot{}v [/mm] $
$ [mm] a=\dot v=\bruch{q\cdot{}p}{m}-\bruch{m\cdot{}g}{m}-\bruch{k}{m}\cdot{}v [/mm] $
$ [mm] m=m_{t}=M-p\cdot{}t [/mm] $
$ [mm] \bruch{dv}{dt}=\bruch{q\cdot{}p}{m(t)}-g-\bruch{k}{m(t)}\cdot{}v [/mm] $
$ [mm] \bruch{dv}{dt}=\bruch{q\cdot{}p}{M-p\cdot{}t}-g-\bruch{k}{M-p\cdot{}t}\cdot{}v [/mm] $
$ [mm] \bruch{dv}{dt}=\bruch{q\cdot{}p-k\cdot{}v}{M-p\cdot{}t}-g [/mm] $
Vielen Dank,
LG, Martinius
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Hallo Martinius,
das sieht richtig aus, aber v=v(t) !
Gruß Christian
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