Hello all. If you will excuse me for thinking philosophically for a moment
about motion, I would like to seek your help in disabusing me of any unknown
assumptions or idiotic notions I am carrying with respect to the following
line of thinking:
Given: Position L and time t, instantaneous velocity v is given by the
derivative of position with respect to time: dL/dt.
The second derivative of L wrt t, dL/dt^2 (equivalently: the derivative of v
wrt t, dv/dt) is instantaneous acceleration a.
The third derivative of L wrt t, dL/dt^3 (equivalently the derivative of a
wrt t, da/dt) is known as jerk - I don't know the proper symbol so I'll call
it j for now.
Assume a situation where an object has v = 0, and a = 0 (any inertial
frame). If the object is to change its inertial state to have a non-zero
final velocity, then the velocity must undergo a change from 0 to some
arbitrary positive value. I recognize that I am assuming that changes in
velocity are smooth and not discontinuous.
If the object is to change its velocity it must have a non-zero value for
acceleration; i.e. to change one's velocity requires an acceleration (either
by changing one's speed or direction). But if an object begins in a
non-accelerated frame where a = 0, in order to do this, the value for
acceleration must change from 0 to some non-zero value...
But this change in acceleration requires a non-zero value for jerk, and so
on ad infinitum.
This seems an unreasonable solution which most likely points to an error in
the analysis, which I hope can be explicitly pointed out to me.
Additionally there is one related problem: that of changes in motion and
whether any arbitrary path can be traveled in a way that keeps jerk to a
value of 0. If jerk = 0 then acceleration must be constant by definition.
But to move along an arbitrary path (the simplest example is circular
motion, say a car driving around in a circle), one cannot keep one's
velocity constant. When acceleration is constant for a car moving on a
circular path, the car can keep jerk =0 and end up driving in a tighter or
wider circle if it changes its direction and velocity in a harmonious way so
that their combined contribution to acceleration = some constant. For
example by driving in a tighter circle, one must apply some amount of
braking to keep overall acceleration constant. Is this the case or am I
missing something obvious? Can jerk be kept to 0 in this instance? Can it
be kept to 0 on any arbitrary path between two points?
Any of your thoughts, comments, insights, or critiques are welcome!