[Phys-L] Re: A problem of motion and derivatives

[Brian Whatcott <betwys1@SBCGLOBAL.NET>]

At 12:16 AM 11/3/2005, Seth Miller, you wrote:

> 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!
>
> Thanks-
>
> -Seth Miller
> East Bay Waldorf School

I offer this, though not quite responsively to your question.

  For people who first go to work doing aircraft simulations,
there comes a moment when they need to model some dynamic
situation.  It often happens that they think in this way:
"I have a variable that describes the position of the dynamic
  object. I can save its value until the next frame (time interval)
and find the difference, and knowing the frame interval,
  I have derived a value for velocity. I could repeat this to
find the acceleration and so derive some force value...."

They are (or ought to be) soon taken in hand, while it is
explained this is a recipe for disaster: the derivatives are
  noisy and compound the errors that stem from variability in
various values.

Instead, they are advised to start with the  most changeable
  integral on hand - which can often be acceleration along
some axis, and integrate this progressively, as needed.

Given acceleration a,
C
C Numerically Integrate
C
v = v (previous) + a * dt
x = x(previous) + v * dt
C
C   Save values until next frame.
C
v(previous) = v
x(previous) = v
    V
    V
    V
The displacement x is forgiving of jitter in higher terms, because
  the change is a modest increment of the total, often enough.

[You will notice this is a simple integration scheme which can easily
  be eleaborated to more accurate approaches - but this is often enough]


Brian Whatcott    Altus OK    Eureka!
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