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.

In[56]:=

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

Out[56]=

  {Cos[t], Sin[t], t}

In[57]:=

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

Out[57]=

  {-Sin[t], Cos[t], 1}

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.

In[58]:=

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

In[59]:=

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

In[60]:=

  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?)

In[61]:=

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

Out[61]=

  Sqrt[2]

In[62]:=

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

Out[62]=

  2 Sqrt[2] Pi + Sqrt[2] t

Out[63]=

  Sqrt[2]

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

In[64]:=

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

Out[64]=

  {-Cos[t], -Sin[t], 0}

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?).

In[65]:=

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

Out[65]=

     Sin[t]    Cos[t]      1
  {-(-------), -------, -------}
     Sqrt[2]   Sqrt[2]  Sqrt[2]

You may have to grab and enlarge the plot again.

In[66]:=

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

In[67]:=

  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