Help Needed Solving A-level Diff Eq Project w/ GPS Data

[Matt1991]

Hi I'm Matt,

I'm new to the Physics Forum and this is my first Post so I'm sorry I don't know how to write out maths on here. I'm doing an A-level differential equations project and I decided to make it a bit more real worldy... maybe a mistake. I am modelling a skydiver in freefall (for the skydivers' velocities) and have got some real GPS data of velocities of the freefall. My project has an initial model and an updated model.

My initial model was using the drag force as 1/2(C)(A)(Rho)v² (keeping the shape factor, the area and the density constant) and it wasn't too difficult really, just rearranging and some integration. The model was clearly wrong though since that model has a terminal velocity but the skydivers' velocities had a slow decrease towards the end and the initial accelaration of my skydivers in the model was not enough.

I decided to make the air density a variable for my updated model. I have modeled this as

Rho = P/(RT) where P is the pressure
T is the temperature
R is the gas constant

The plan originally was to solve the final equation using the finite element method for solving second order differentials and then use the central difference method for finding the velocities. Unfortunately I have found that very hard as my textbook has no notes on dealing with equations with (ds/dt)ⁿ. It also only shows examples when s, s' and s'' are in separate terms rather than multiplied together.

Any help solving this would be greatly greatly appreciated (and sorry for the huge amount of text)

The Equation I am working with: (any thing dashed e.g. v' is differentiated with respect to t)(b is a constant, as is D)

mv' = mg - (1/2)(C)(A)(P₀/(RT₀))e^bsv²

mv' = mg - De^bsv

where D = (1/2)(C)(A)(P₀/(RT₀))

Thanks,
Matt

 

[coomast]

Hello Matt1991,

Welcome to the forum!

If you need help on typing the math, have a look at this document:

https://www.physicsforums.com/misc/howtolatex.pdf" [Broken]

Now rewriting you're equation as:

$$\frac{dz}{dt}=g-K\cdot e^{bt}\cdot z^2=g$$

It is identified as a form of the Riccati differential equation and this is not easy to solve. Only in a limited number of cases it is possible. Unfortunaltely (I think) not here, so there is not much we can do, except try using a numerical approach like Runge-Kutta, or perhaps a series solution. The latter is something I have not tried yet, so I can't help you any further on this.

I solved this equation in another post, but this one was simple enough to integrate. Just for the info, this is the link:

https://www.physicsforums.com/showthread.php?p=1603888#post1603888"

hope this helps a bit,

coomast

 

[Matt1991]

Thanks for the help. I figured that I wouldn't be able to solve this analytically and looked through the numerical solutions which is where I decided to try the Finite Element Analysis followed by the central difference method since finte element would give me a set of displacements and the central difference then the velocities (order of error in methods not being a big issue here). Theres where I got stuck, I really could not see how to apply the finite element analysis to the problem and was wondering if it is possible and if there is another numerical method that can be applied to this equation.

Thanks for the links too

Matt

 

[Matt1991]

In case anybody wants to know I did solve this for v and t.

i rearanged using dv/dt = v.dv/ds

I ended up with dv/ds = (g/v) - (D/m)(e^bs)(v)

I then used the Improved Euler Method Using Runge Kutta notation for ease to solve for v and s. I then used this information to create a table of values for v and t.

Easy enough really, just did not see it at all for a while. Solved my own problem :D

Matt