[RyanCahoon]

Originally Posted by Ether

5a) What are the XY coordinates 30 seconds later?

Spoiler for 5a:

T=0: (0,0) pi/12 (from +Y axis)

F(t) = 5*sin(t/2)
S(t) = 4*sin(t/2.2)
dQ = 1.5*sin(t/2.5)

Q(t) = (Integrate[1.5*Sin[t/2.5], t, 0, t]) + Pi/12 = 4.0118-3.75*Cos[0.4*t]

T=30:
dX(t) = F(t)*Sin[Q(t)]+S(t)*Cos[Q(t)]
X = Integrate[dX(t), t, 0, 30]
= Integrate[5*Sin[t/2]*Sin[4.0118-3.75*Cos[0.4*t]] + 4*Sin[t/2.2]*Cos[4.0118-3.75*Cos[0.4*t]], t, 0, 30]
= -5.94429

dY(t) = F(t)*Cos[Q(t)]-S(t)*Sin[Q(t)]
y = Integrate[dY(t), t, 0, 30]
= Integrate[5*Sin[t/2]*Cos[4.0118-3.75*Cos[0.4*t]] - 4*Sin[t/2.2]*Sin[4.0118-3.75*Cos[0.4*t]], t, 0, 30]
= -9.11027

Originally Posted by Ether

5b) What is the path length traveled during that 30 seconds?

Spoiler for 5b:

T=0: (0,0) pi/12 (from +Y axis)

FWD = 5*sin(t/2) ft/s
STR = 4*sin(t/2.2) ft/sec
RCW = 1.5*sin(t/2.5) radians/sec

T=30:
distance = Integrate[Sqrt[(dX(t))^2+(dY(t))^2], t, 0, 30]
= Integrate[Sqrt[(5*Sin[t/2]*Sin[4.0118-3.75*Cos[0.4*t]] + 4*Sin[t/2.2]*Cos[4.0118-3.75*Cos[0.4*t]])^2+(5*Sin[t/2]*Cos[4.0118-3.75*Cos[0.4*t]] - 4*Sin[t/2.2]*Sin[4.0118-3.75*Cos[0.4*t]])^2], t, 0, 30]
= 129.597