Fall with Air Drag - Which Value for Velocity to Use?

[dracayr]

Homework Statement

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):

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 v_n, so the curve would always be a bit lower than 1/2at². So I fixed that by changing the Equation to

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 v_n and not (v_n+v_n+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 (v_n+v_n+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

F_d=b|v|^m
v_E=(mg/b)^1/m
a=g(1-v^m/v_E^m)

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

The Attempt at a Solution

These are the spreadsheets with the 2 systems:

http://dracayr.awardspace.com/physik.ods"
http://dracayr.awardspace.com/physik.xls"


dracayr

 

[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.

 

[dracayr]

OK, thanks for the information :)

dracayr