Homework Help: Freefall DE Model

1. Mar 14, 2014

Big Triece

1. The problem statement, all variables and given/known data

One of my classes involves constructing a differential equation to model the freefall of the red bull sky diver Felix Baumgartner. I need to construct the correct differential equation to find v(t) and y(t) at any given time t. As stated above, I need to take the two forces (drag and gravity) as varying with height. I'm simply interested in how to construct the model, I can worry about solving it on my own.

2. Relevant equations

ma = $F_{g}$ - $F_{d}$

v' = vdv/dy

3. The attempt at a solution

ma = mvdv/dy = GMm/(R + $y^{2}$) - 1/2$C_{d}$A$\rho$$v^{2}$

where R, G, M, and m are the usual gravitational constants, $C_{d}$ is the drag coefficient, A is the cross sectional area of the diver, and rho is the density of the air.

I'm a little perplexed because I believe rho should be a function of y as well. I was wondering if I should just treat the drag force as 1/2k$v^{2}$ and solve accordingly. I also might need to add the linear term for the drag force although it gets dominated once v gets larger. Any thoughts as to how bad I butchered this model are appreciated.

Thanks

2. Mar 14, 2014

Staff: Mentor

For your application, the effect of altitude on gravitational acceleration is going to be negligible, so you might as well use the value at the surface. You can take into account the effect of altitude on density pretty easily.

The hard part is going to be quantifying the drag coefficient. There are correlations for Cd as a function of the Reynolds number in the literature for specific shapes, but you need to find it for your shape (i.e., the shape of a human body). Also, the orientation of the person's body is going to affect the drag coefficient (whether he is in sky diver orientation or with feet straight down, or tumbling). The drag coefficient will vary strongly with the orientation, and so also will the projected area of the object. Finding the data you need on this is really going to be the key complexity in executing this project, and also will be key in making accurate predictions with you model.

Chet