Computer Algebra Recipes

180 CHAPTER 4. NONLINEAR ODE MODELS
As she was taught way back in ¯rst-year physics, Jennifer begins by making
a free-body diagram showing all the forces acting on the golf ball. This diagram
is reproduced in Figure 4.9.
φ
golf ball
F
F F
drag
lift
θ
V
gravity
Figure 4.9: Free-body diagram for a golf ball, including drag and lift.
Three forces are included, the pull of gravity ~ Fgravity downward, the drag force
~Fdrag in the direction opposite to the instantaneous velocity vector ~ V , and
the lift force ~ Flift in a direction perpendicular to ~ V . For simplicity, Jennifer
assumes that the ball's spin axis is in the z-direction and that the translational
motion is in the x-y plane with x horizontal and y vertical. She makes a call to
the plots and VectorCalculus packages,
> restart: with(plots): with(VectorCalculus):
and enters the velocity vector ~ V = Vx ^ex + Vy ^ey.
> Velocity:=;
Velocity := Vx ex + Vy ey
Since the speed V will be needed, it is calculated by taking the dot product of
the velocity vector with itself and then taking the square root. The dot product
is entered with the shorthand3 dot syntax. The unit vector (assigned the name
uv) in the direction of ~ V is then equal to the velocity divided by the speed.
> Speed:=sqrt(Velocity . Velocity); u[v]:=Velocity/Speed;
uv := q
q
Speed :=
Vx
Vx 2 + Vy 2
ex +
V
x
Vx 2 + Vy 2
q
3 The long form is DotProduct(Velocity,Velocity).
V
y
Vy
Vx 2 + Vy 2
ey