Velocity and Acceleration

If a particle is moving so that at time t it's position is
F(t)=(x(t), y(t), z(t)),
then its velocity is given by
v(t)=F'(t)=(x'(t), y'(t), z'(t)).
The velocity is a tangent vector to the curve of motion (unless v(t)=0). At any instant, the particle is moving in the direction of the tangent.


  F[t_]:={Cos[t],Sin[t],t};
  F[t]


  v[t_]:=F'[t]
  v[t]

To get a better look at the graphs of F[t] and v[t], click on the plot below and drag one of the black squares with the mouse to enlarge it. You should see very easily the tangent to the curve.


  motion=ParametricPlot3D[F[t],{t,-2 Pi,2 Pi}];


  tang=vectorPlot[Table[F[i],{i,-4,4}],Table[F'[i]+F[i],{i,-4,4}]];


  Show[motion,tang,AspectRatio->Automatic];

The speed is given by the magnitude of the velocity,
||v(t)||=||F'(t)||.
Note that since the distance function is
s(t)=Integral from a to t of ||F'(w)||dw,
the speed is also s'(t). (Do you see why?)


  speed[t_]:=norm[v[t]]
  Simplify[speed[t]]


  s[t_]=arclength[F,-2 Pi,t]
  s'[t]

The acceleration is the rate of change of the velocity,
a(t)=v'(t)=F''(t)


  a[t_]:=F''[t]
  a[t]

The unit tangent vector, T(t), is simply T(t)=F'(t)/||F'(t)|| or T(t)=F'(t)/s'(t) (do you see why?).


  T[t_]:=F'[t]/Sqrt[F'[t].F'[t]]
  Simplify[T[t]]

You may have to grab and enlarge the plot again.


  unittang=vectorPlot[Table[F[i],{i,-4,4}],Table[T[i]+F[i],{i,-4,4}]];


  Show[motion, unittang,AspectRatio->Automatic];

Let's see it again in two dimensions.

Let's see it again in two dimensions. ...

Up to Vector Differential Calculus