Computer Algebra Recipes

4.2. SECOND-ORDER MODELS 181
Referring to the freebody diagram, sin(μ) =Vy=V and cos(μ) =Vx=V .
> sin(theta):=V[y]/Speed; cos(theta):=V[x]/Speed;
sin(μ) := q
Vy
Vx 2 + Vy 2
cos(μ) := q
Vx
Vx 2 + Vy 2
The unit vector ^ Á pointing in the lift direction is related to the Cartesian unit
vectors ^ex and ^ey pointing along the x- andy-directions, respectively, by the
relation ^ Á = ¡ sin μ ^ex +cosμ ^ey. This relation is entered and labeled uÁ.
> u[phi]:=;
uÁ := ¡ q
Vy
Vx 2 + Vy 2
ex +
q
Vx
Vx 2 + Vy 2
ey
The gravitational force ~ Fgravity = ¡mg^ey, wherem is the mass of the ball and
g is the acceleration due to gravity.
> F[gravity]:=;
Fgravity := ¡mgey
According to the photocopied references, both the drag force ~ Fdrag and the
lift force ~ Flift are proportional to the square of the speed. Jennifer labels the
proportionality constants (per unit mass) as Kdrag and Klift, respectively, and
enters the two forces.
> F[drag]:=-K[drag]*m*Speed^2*u[v];
q
Fdrag := ¡Kdrag m
Vx 2 + Vy 2 Vx ex ¡ Kdrag m
q
Vx 2 + Vy 2 Vy ey
> F[lift]:=K[lift]*m*Speed^2*u[phi];
q
Flift := ¡Klift m Vx 2 + Vy 2 q
Vy ex + Klift m Vx 2 + Vy 2 Vx ey
The net force ~ F acting on the golf ball is the vector sum of the three forces.
> F:=F[gravity]+F[drag]+F[lift]:
Now, Vx = dx(t)=dt and Vy = dy(t)=dt, wherex(t) andy(t) are the horizontal
and vertical coordinates of the golf ball at time t.
> V[x]:=diff(x(t),t): V[y]:=diff(y(t),t):
Newton's second law of motion is applied in the x- andy-directions.
> xeq:=diff(V[x],t)=simplify(F[1]/m);
xeq := d2
s
μ 2 μ
d d
x(t) =¡ x (t) +
dt2 dt dt y(t)
2μ μ μ
d
d
Kdrag x (t) + Klift
dt dt y(t)
> yeq:=diff(V[y],t)=simplify(F[2]/m);