User:EGM6341.s11.TEAM1.WILKS/Mtg23

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_10_12_15_02_11.djvu | Mtg 23:]] Tue, 12 Oct 10

= Lecture plan, contents, servers =

[[media: 2010_10_12_15_02_11.djvu | Page 23-1]]
-Lecture plan, table of contents: Course organization; References specific to sections F09 Apollo server: http://clesm.mae.ufl.edu/~vql/courses/pea1/2009.fall/ clesm.ippd: http://clesm.ippd.ufl.edu/~vql/courses/pea1/2009.fall/

= Soccer physics =

NOTE: Soccer physics: sport projectiles $$ \Rightarrow \ \ $$ must account for air resistance: [[media: 2010_09_21_14_56_48.djvu | see Mtg 13]]

Drag force: $$ F_D= \rho\ \frac{U^2}{2}AC_D \ $$ $$ \rho\ = \ $$ air mass density $$ A = \ $$ area of cross section $$ C_D = \ $$ drag coefficient



[[media: 2010_10_12_15_02_11.djvu | Page 23-2]]
$$ D = \ $$ diameter of ball $$ \nu\ = \ $$ kinematic viscosity $$ \mu\ = \ $$ dynamic/kinetic viscosity

END NOTE

= Exact Nn-ODEs [[media: 2010_10_08_17_07_22.djvu | p.22.5]] cont'd =

Case $$ n = 2 \ $$ : N2_ODE

[[media: 2010_10_12_15_02_11.djvu | Page 23-3]]
Eq.(1)p.23-2 is equivalent to 2 relationships for exactness of condition 2 for N2-ODEs: See F09 [[media: Egm6321.f09.mtg13.djvu | Mtg 13]] [[media: Egm6321.f09.mtg14.djvu | Mtg 14]] for the details on the homework problem statement. Below is a detailed explanation.

where $$ p=y' \ $$ $$ q=y'' \ $$ $$ \phi_x + \phi_y p= g(x,y,p) \ $$ $$ \phi_p = f(x,y,p) \ $$

where $$ q=y'' \ $$

where $$ q=y'' \ $$

[[media: 2010_10_12_15_02_11.djvu | Eq.(1)p.23-2]] $$ \Rightarrow \ \ $$

Define

$$\bar g := f_{xx}+2pf_{xy}+p^2f_{yy}-g_{px}-pg_{py}+g_y \ $$

$$\bar f := (f_{xp}+pf_{yp}+2f_y-g_{pp})\ $$

[[media: 2010_10_12_15_02_11.djvu | Page 23-4]]
[[media: 2010_10_12_15_02_11.djvu | Eq.(6)p.23-3]] can be rewritten as

$$ \bar g \cdot 1 + \bar f \cdot q = 0 \ $$

For [[media: 2010_10_12_15_02_11.djvu | Eq.(6)p.23-3]] to hold, the independence of $$ 1 $$ and $$ q $$ (since $$ q = y'' $$ is not a constant in general) implies that $$ \bar g =0 \ $$ and $$ \bar f =0 \ $$.

Recall $$ \phi\ (x,y,p) \ $$