I spring the body electric

So:

This is the laTex for the equation t = \sqrt{\frac{2d}{a}}

Since v=at then average v is v_f / 2:

So v=at and thus:

And this just seems backward to me, trying to look into a mirror and hunt squirrels.

As anyone who reads the blog regularly knows I wander aimlessly through a vast landscape of concepts daily and tonight it is harmonic motion. These are some equations of Latex derivation. ( I love my Latex and my SpeedCrunch ).

It seems to me at first glance that since average v and sqrt(d) are proportionate that it only follows that it will be a constant at a specific height if you assume an ideal case without friction and air resistance. By forcing the path to be longer than a simple harmonic , you alter the time period. I have been delving into the history of mathematics for about 2 hours and it seems that many of these things are considered "intractable" and I don't see that at all.

It follows that since Gravity ( variable a, here ) is assumed to be constant and is generally so, and t = d / v and d is proportionate to v² then t is proportionate to v which is only determined by the length of the oscillation. ( now two interdependent constants ) then, the relationship of d : v : t is inherently constant by simple substitution.

In any case I am learning some interesting relationships about the origin of the mathematics. This really started at curve fitting and the least squares rule and somehow I got to this, for the moment, and from there it was in Python to do a Fourier of something that was intended to be a mathematical joke, however it has taken too long to fabricate. It is interesting and I can see I have much to learn yet ( always ).

I remember my real point in all this, by measuring the period of a pendulum over time it is possible to know "G" more and more precisely in a specific location and I think that has some significance.

Double pendulums and chaos are also very interesting.

I see there is more for me to learn here too as the interaction of the forces can have some unusual effects when acting in a rotating system and I suppose that is what Focault is all about. I never fully grasped the interactions of those forces in an intuitive way and I think that I might be able to do that now. The study of matrix math, openGl and now blender has vastly improved my ability to visualize and understand relationships that had been accepted, because they are measurable. I think there are some unusual applications that should be possible if the underlying principle of system is understood. Gyroscopes are an interesting subject and the way in which they operate seems counter intuitive to the idea of steady state gravity. It does have some influence on the leverage against Gravity and I will likely study that tomorrow. I suspect that buried in those relationships is a time-rate t³ of action variance in momentum.

ABOUT ERRORS: I made a mistake when performing the transfer of "a" to the equation to isolate "t". So I made the corrections now and deleted the rest as it would just be confusing.