trigonometric vector equation

rikoshea_at_yahoo.com
Date: 03/22/05 

Date: Tue, 22 Mar 2005 13:46:29 +0000 (UTC)

Hello,
I hope I've identified the right forum for my question.

I have a problem understanding an apparent annomaly related to a
vector equation for calculating apparent wind velocity associated with
bicycle motion. Maybe someone could explain what's going wrong.

This link displays a diagram:
http://www.sheldonbrown.com/brandt/wind.html

W = sqrt((U+V*cos(a))^2 + (V*sin(a))^2)

Where:
U = bicycle velocity
V = wind velocity
a = angle of wind (0 = head-on, 180 = tailwind)
W = apparent wind velocity

The above equation could also be written using
the law of cosines as: W^2 = U^2 + V^2 - 2.U.V.cosW

The problem I encounter in when the value V is greater than U for
obtuse angle of a, i.e. when there is a tailwind and the wind velocity
is greater than the bicycle velocity.

As an example:
Case (1)

U = 20
V = 19
a = 180 degree (direct tailwind)
cos(180) = -1
sin(180) = 0

W = sqrt((U+V*cos(a))^2 + (V*sin(a))^2)

W = sqrt((20+19*-1)^2 + (19*0)^2)
W = sqrt((1)^2)
W = 1
This is ok, the resulting wind velocity is correct.

Case (2)
This is the problem case when V is greater than U.
U = 20
V = 21
a = 180 degree
cos(180) = -1
sin(180) = 0

W = sqrt((U+V*cos(a))^2 + (V*sin(a))^2)

W = sqrt((20+21*-1)^2 + (21*0)^2)
W = sqrt((-1)^2)
W = 1
This is incorrect, as I would have expected the resulting wind velocity
to be -1.

Can any one throw any light on this problem? Should a different
equation be used ?

Thanks in advance