Lecture Oct 29 -- Curves, acceleration, curvature, etc.

Velocity, tangent vector and acceleration

Clear[X, V, A, T, speed, Atang, Anormal]
X[t_] := {Cos[(t^3)], Sin[(t^3)], (t^3)}

<< Default3D`
ParametricPlot3D[X[t],  {t,  -2,  2}]

norm[v_]  :=  Sqrt[v . v]

V[t_] = (∂_t) X[t]
A[t_] = Simplify[(∂_t) V[t]]
T[t_] = Simplify[V[t]/norm[V[t]]]

speed[t_]  =  Simplify[norm[V[t]]]

Remark:  This is not constant speed motion!  Therefore we expect non-zero tangential  acceleration.

Atang[t_]  =  
      Simplify[A[t] . T[t]]  T[t]
Atang[t]  ==  D[speed[t],  t] T[t]

Anormal[t_]  =  A[t]  -  Atang[t]

Curvature

curv[t_]  =  Simplify[D[T[t],  t]/speed[t]]

kappa[t_]  =  Simplify[norm[curv[t]]]

normal[t_] =  Simplify[curv[t]/norm[curv[t]]]