Re: da/dt = jerk?

[Hugh Haskell <hhaskell@MINDSPRING.COM>]

Guentcho's remark about how rails are laid reminds me of a similar
observation I made many years ago when I got my son his first
electric train, but didn't put it into this context until just now.
When assembling the tracks, the curved pieces fit nto the straight
pieces in such a way that the straight piece is tangent to the circle
that the curved piece is an arc of. It would seem that this should
allow the motion of the train on its circuit around the track to be
smooth. But looking at the train from above the locomotive has a
noticeable jerk as it enters the curve and suddenly finds itself
being centripetally accelerated. You can actually see the train rock
a bit as it enters the turn, and the turning motion starts suddenly,
not smoothly, as you might expect. The same thing happens as it comes
out of the curve. The turning motion stops suddenly and the train
rocks a bit. It makes sense when you think that the radius of
curvature went suddenly from that of the circle to infinity (or vice
versa). There is clearly a large and sudden change in acceleration.

This may also be part of why the curvature of the vertical loops in
amusement park roller- coasters. It is also true that the curvature
is smaller at the top so that the force on the car is more nearly
uniform as it slows down on the way up and speeds up on the way down,
but it looks like the entry into the curve is by gradually reducing
the radius of curvature from infinity down to the value at the top.
Of course there has to be some discontinuity from infinity to the
initial value unless we want the loop to take infinite time to
negotiate, but if it the initial curvature is small (large radius of
curvature) the jerk is minimized at the beginning, and the force
build-up can be made not too uncomfortable, and then held nearly
constant until it's time to come out of the curve.

During my years as a Naval Aviator, I spent some time flying
transport aircraft, and I recall that we spent some time practicing
stopping the aircraft so that the passengers would not realize we
were stopped (judging from my recent experiences on airliners, this
skill is a lost art). We did it by gradually slowing down as we
approached the parking spot and then applying the brakes very gently.
Even when it was a "good" stop there was usually some jerk as the
acceleration went to zero. But every so often we got one in which the
acceleration and the velocity got to zero simultaneously, and the
passengers would sit there for two or three seconds, not realizing
that we had stopped. Of course it was also possible to make smooth
landings in which the downward component of velocity got to zero just
as the wheels touched the ground, but there were too many other
things going on there for the landing to go unnoticed (wheels
starting turning as they roled along the ground, nose wheel dropping
onto the ground, brakes and thrust reversal being applied heavily--it
wouldn't do to make a "greaser" landing and then run off the end of
the runway). Even so, these events were much rarer than the smooth
stops.

We will be talking about circular motion in my classes soon. I now
have lots more things to talk about in that context regarding jerk
than I used to have, thanks to this thread.

Hugh

> BTW: I have read in an analysis book that rail bends have are not just parts
> of circles, but other curves, because coming into a circle would mean a
> sudden change in the normal component of acceleration. The author's idea was
> that, although the speed doesn't change with a step, a sudden change in the
> acceleration (appearance of a force) is, at least, unpleasant (is that
> really a problem? I mean, the maximum value of the force/acceleration is the
> same anyway, well, if the bend is long enough; or is the sudden change "bad"
> enough, e.g. because of the wide spectrum?).
>
> Bye
> Guentcho


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Hugh Haskell

<mailto://haskell@odie.ncssm.edu>
<mailto://hhaskell@mindspring.com>

The box said "Requires Windows 95 or better." So I bought a Macintosh.
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