## General Kinematic Theorem

Previous section: Circular Motion Model.

Next sectionGeneral Keplerian Motion.

x = x(s) = parametric equation for particle path.

x
= x(t) = parametric equation for particle trajectory

Path length:
s = s(t)

Velocity:

$\Large{\frac{{d{\bf{x}}}}{{dt}}\,\,\, = \,\,\,{\bf{v}}\,\,\, = \,\,\,{\bf{\hat v}}v\,\,\, = \,\,\,\frac{{d{\bf{x}}}}{{ds}}\,\,\frac{{ds}}{{dt}}}$

Speed:  ds / dt

Therefore, tangent direction:

$\Large{{\bf{\hat v}}\,\,\, = \,\,\,\frac{{d{\bf{x}}}}{{ds}}\,\,\, = \,\,\,{e^{{\bf{i}}\theta }}\,{{\bf{\hat v}}_0}}$

This summary is generally valid for position histories in any dimension. We can always say the current tangent direction can be thought of as a rotation of the initial tangent direction (or of any reference direction). And, the rotation plane and rotation amount can always be represented by some unit bivector i and some angle θ. But, since this GA summary has been concerned mainly with two dimensional applications, restricting ourselves to a fixed plane, we will consider the unit bivector i to be constant for the theorems below.

Theorems:

Velocity:  v
is tangent to the particle path (trajectory).

Acceleration:

$\Large{{\bf{a}}\,\,\, \equiv \,\,\,\frac{{d{\bf{v}}}}{{dt}}\,\,\, = \,\,\,\frac{d}{{dt}}\left( {{\bf{\hat v}}v} \right)\,\,\, = \,\,\,{\bf{\hat v}}\frac{{dv}}{{dt}} + {\bf{iv}}\frac{{d\theta }}{{dt}}\,\,\, = \,\,\,\left( {\frac{{dv}}{{dt}} \pm {\bf{i}}\frac{{{v^2}}}{r}} \right){\bf{\hat v}}}$

points inside the curving path and the radius r of the osculating circle
is defined by  v = ds / d= r |dθ/dt|.

Exercise: A particle slides on the frictionless track below subject to a force  mg + N = m dv / dt  that keeps it on the track. Sketch its velocity v and acceleration a at the indicated points: