- #1
MaceLee
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Hi everyone,
I've been having a bit of trouble with my mathematics coursework. We have to model an aeroplane that has just landed with two equations. One prior to braking, one after. I've used a model with a quadratic drag approximation; and I'm not certain if my result is correct. Mainly because we haven't covered anything to do with non-linear differential equations, and suddenly I've found myself with one.
I'm modelling the system before a constant braking force with this DE:
[tex]m\ddot{x} = -k\dot{x}^2[/tex]
I'm trying to find [tex]\ddot{x}[/tex], but I'll probably need to find [tex]x[/tex] as well
[tex]v = \dot{x}[/tex]
[tex]m\dot{v} = -kv^2[/tex]
[tex]\dot{v} = -(k/m)v^2[/tex]
[tex]-(\frac{m}{k})\int\frac{1}{v^2} dv = \int dt[/tex]
[tex]\frac{m}{kv} + c = t[/tex]
[tex]\dot{x} = v = \frac{m}{k(t-c)}[/tex]
Is this correct? All the constants I'll find by trial and error matching it up to the velocity-time data we were given.
Thanks :)
Homework Statement
I've been having a bit of trouble with my mathematics coursework. We have to model an aeroplane that has just landed with two equations. One prior to braking, one after. I've used a model with a quadratic drag approximation; and I'm not certain if my result is correct. Mainly because we haven't covered anything to do with non-linear differential equations, and suddenly I've found myself with one.
Homework Equations
I'm modelling the system before a constant braking force with this DE:
[tex]m\ddot{x} = -k\dot{x}^2[/tex]
I'm trying to find [tex]\ddot{x}[/tex], but I'll probably need to find [tex]x[/tex] as well
The Attempt at a Solution
[tex]v = \dot{x}[/tex]
[tex]m\dot{v} = -kv^2[/tex]
[tex]\dot{v} = -(k/m)v^2[/tex]
[tex]-(\frac{m}{k})\int\frac{1}{v^2} dv = \int dt[/tex]
[tex]\frac{m}{kv} + c = t[/tex]
[tex]\dot{x} = v = \frac{m}{k(t-c)}[/tex]
Is this correct? All the constants I'll find by trial and error matching it up to the velocity-time data we were given.
Thanks :)
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