Differential Equation Problem 2

**eXist** · September 24th 2009, 05:45 PM

Once again a friend has a diff-e-q problem that he needs help with:

A woman bails out of an airplane at an altitude of 10,000 ft, falls freely for 20 s, then opens her parachute. How long will it take her to reach the ground? Assume linear air resistance v ft/s2, taking $\rho = 0.15$ without the parachute and $\rho = 1.5$ with the parachute.

**chisigma** · September 24th 2009, 11:33 PM

If you suppose that at the time $t=0$ the speed along the x axis is null [unfortunately that isn't...] the problem is onedimensional and the 'move equation' is...

$y^{''} = - g + \rho\cdot y^{' 2}$ , $y(0)=h$ , $y^{'} (0) = u_{0}$ (1)

Setting $y^{'} (t) = u(t)$ the (1) becomes...

$\frac {du}{dt} = - g + \rho\cdot u^{2}$ (2)

... that is more 'tractable' if you swap the u with the t so that...

$\frac{dt}{du} = - \frac{1} {g - \rho\cdot u^{2}}$ (3)

Integrating both terms of (3) you obtain...

$t= - \frac{1}{\sqrt{\rho g}}\cdot \tanh^{-1} \sqrt{\frac{\rho}{g}}\cdot u + c$ (4)

... and from it...

$u= -\sqrt{\frac{g}{\rho}} \tanh \sqrt{\rho \cdot g} \cdot (t+c_{1})$ (5)

Integrating both terms of (5) and taking into account that is...

$\int \tanh (ax +b) dx = \frac{1}{a} \cdot \ln \cosh (ax + b) + c$ (6)

... we obtain finally...

$y= -\frac{\ln \cosh (\sqrt{\rho\cdot g}\cdot t + c_{1})}{\rho} + c_{2}$ (7)

At this point you have to solve two distinct problems...

a) first you solve the (1) for $0 \le t \le 20 s$ with $\rho= .15$ and initial conditions $y(0)=h$ , $y^{'}(0)=0$ ...

b) second you solve (1) for $t>20 s$ with $\rho=1.5$ and initial conditions determined by the step a)...

Kind regards

$\chi$ $\sigma$

**chisigma** · September 28th 2009, 12:31 AM

In the problem proposed by eXist we have hypothesized that the aircraft’s speed is negligible, so that the parachute jumper’s trajectory is one dimensional. Of course in real situations that isn’t, so that I propose You an example in the second world war scenario…

Target of the B-17 bomber is a German radar mobile station the destruction of which is necessary for the success of the D-Day operations. The pilot of the B-17 has to find the time or alternatively the distance $d$ from the target at which to drop the bomb load. He knows…

a) the [constant] altitude of flight $h$ and the horizontal speed $v_{0}$ of the B-17…
b) the aerodinamic coefficient $\rho$ of the bombs…

How to solve his problem and find $d$ ?…

Kind regards

$\chi$ $\sigma$

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