Fall with air drag

1. Feb 13, 2009

dracayr

1. The problem statement, all variables and given/known data
I want to numerically simulate a fall with air drag. In class, we used these equations to produce a table simulating free fall (using a ti calculator):
Code (Text):
dt=0.1s
a=9.81m/s²
t[SUB]n+1[/SUB]=t[SUB]n[/SUB]+dt
v[SUB]n+1[/SUB]=v[SUB]n[/SUB]+a*dt
x[SUB]n+1[/SUB]=x[SUB]n[/SUB]+v[SUB]n[/SUB]*dt
When I compared that system with the curve produced by the direct equation, x=1/2at², I noticed a slight error. After looking at the equations again, I figured that the error came from the fact that I used vn, so the curve would always be a bit lower than 1/2at². So I fixed that by changing the Equation to
Code (Text):
x[SUB]n+1[/SUB]=x[SUB]n[/SUB]+((v[SUB]n+1[/SUB]+v[SUB]n[/SUB])/2)*dt
Now I want to create a similar system but with air drag. As before, my physics book says that I should use vn and not (vn+vn+1)/2. But the book was wrong when it came to free fall, so I don't know which value to use for v. I think I should use (vn+vn+1)/2. But since I don't know the direct equation this time (if there is one), I can't compare to the correct solution..

2a. Relevant equations
v=x/t
a=v/t
F=ma

Fd=b|v|m
vE=(mg/b)1/m
a=g(1-vm/vEm)

Fd is the air drag
b and m are constants depending on the drag (I think in the case of air, m is about 2)
vE is the end velocity the falling body will achieve

3. The attempt at a solution

These are the spreadsheets with the 2 systems:

http://dracayr.awardspace.com/physik.ods" [Broken]
http://dracayr.awardspace.com/physik.xls" [Broken]

dracayr

Last edited by a moderator: May 4, 2017
2. Feb 13, 2009

minger

Ah, you've unknowingly ran into problems solving differential equations numerically. Essentially, what you're doing is time-marching an ordinary differential equation. You're comparing results using upwind and central differencing. For what you're doing, just do as the book says.

Numerically, upwinding (essentially using the value of the velocity at the current time) will damp waves, so you will typically have a lower magnitude than the analytic solution.

3. Feb 14, 2009

dracayr

OK, thanks for the information :)

dracayr

Last edited: Feb 14, 2009