velocity vector v(t) is defined as
The speed of a particle at time t is given by the magnitude of the velocity vector:
and hence is always positive.
acceleration vector a(t) is defined as
Newton's Second Law of Motion:
centripetal force: a force acting along the vector r(t) toward the origin. A ``center-seeking'' force, like gravity.
Acceleration can be broken into two components: one in the direction of the tangent vector, and the other in the direction of the normal vector:
and using the definition of curvature and N, we obtain
with tangential components
Alternatively,
None to speak of.
First of all, velocity and acceleration are simple generalizations of the laws for univariate motion.
As a particle moves through space, its motion is always in the plane defined by T and N; these two vectors constitute a basis for the osculating plane. The two components of acceleration are in the directions of T and N: tangential acceleration is driven by a change in speed, whereas the normal component is driven by a change in curvature of the path.
In terms of a car's motion, hitting the accelerator on a straight-away pushes you back against your seat: this is tangential acceleration; going around a corner is normal acceration, thrusting you towards your door, or your significant other in the seat beside you. Another type of normal acceleration is the so-called ``tummy-tickler'': if you go over a rise in the road at a constant speed, you get a thrill in your tummy, which is all about normal acceleration.
With the ammunition which we have in this section, we can derive Kepler's first law of planetary motion from Newton's Second Law of Motion and his Law of Universal Gravitation. The derivation is a little ugly, but you will see that it is within your grasp. There are, however, a few little tricks in the derivation, which indicate that it might have taken you and me a little while to figure out how to get the answer!