In physical (especially mechanics) problems we often have solutions in a form , a “vector function”.
Example 4.12:
A particle moves along a circle with uniform angular frequency, . Find the velocity.
Solution:
If we are perfectly naive, we write . This is actually correct!
The velocity is defined as the vector with as components the time-derivative of the components of the position vector,
It is actually quite illustrative to look at a graphical representation of the procedure, see Fig. ??. We notice there that the (vector) derivative of a vector function points is a vector that is tangent to (describes the local direction of) the curve: not a surprise since that is what velocity is!
Example 4.13:
When a particle moves in a circle, find two independent way to show that .
Solution:
1) Use the uniform motion example from
above, and we find
. This is not a general answer though!
2) Write
. (Definition of circle!) Then, by
differentiating both sides of the relation (in the
“other” order), we find
and we have the desired results.
Example 4.14:
Find the velocity of a particle that moves from to in along a straight line with constant velocity. Also find the position after passing ,
Solution:
Clearly if the particle is at point 1 at , We get, substituting ;
from which we conclude (solving for each component separately) that . At time we have
Things get slightly more involved (but quite relevant!) when we look at curves in polar coordinates, i.e., specified by and . From we find that
The first unit vector is indeed the one parallel to ; the second one is defined from its expression. There is some interesting mathematics going on over here,
This is often used to say that and are orthogonal coordinates, at each point they are associated with different, but always orthogonal directions!
Example 4.15:
Express the velocity of a particle moving in an elliptic (Kepler) orbit,
in turn of . Now calculate the kinetic energy of the particle.
Solution:
Obviously . Now differentiate w.r.t. using the chain and quotient rules:
The kinetic energy is thus found to be