Ok. Now to get into some horrible math notation as promised.
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Let's define some notation and definitions:
X(t) represents a position vector.
I will denote the position vector of an object p, which is a function of time, as X_p(t).
(So X_p(t) = (x_p(t), y_p(t), z_p(t)) in cartesian coordinates)
f'(t), f''(t), ... represents the first derivative of a function f with respect to t, the second derivative of f wrt t, etc
S[a,b]f(t)dt represents the definite integral of the function f(t) wrt to t,with lower limit t=a, upper limit t=b
(S was the closest thing I saw on the keyboard that looked kind of like the integral sign)
What i mean by two things having the same motion is: their velocity graphs are the same and their position graphs vary at most by a uniform translation. You could probably come up with a more general def. for motion, but this works for here. (for instance maybe make it rotationally invariant also)
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Once again. My claims:
<BLOCKQUOTE id=quote><font size=1 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>
Proposition 1: Given two objects, possibly of different mass, denote their masses as m1 and m2 respectively. Assuming both object's center of mass experience the same accelration, denoted by the function a(t), and have the same initial velocity, X'(0), then each's center of mass ,X_p1(t) & X_p2(t), will follow the exact same motion.<hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
(Note, I could have claimed some thing about object orientation, etc but I tried to keep things as simple as possible. The conclusion would be the same anways. I actually already made it more general than is needed, can you can spot it?)
<BLOCKQUOTE id=quote><font size=1 face="Verdana, Arial, Helvetica" id=quote>quote:<hr height=1 noshade id=quote>Corollary 1: If two objects have the properties given in claim 1, then they will have the same turning radius. <hr height=1 noshade id=quote></BLOCKQUOTE id=quote></font id=quote><font face="Verdana, Arial, Helvetica" size=2 id=quote>
This immediately follows Prop. 1 since both follow the same motion, then, regardless of mass, their turning radius will be the same.
(Notice there is no talk of forces in here. We can disregard all the aerodynamics, mass, smell, color, whatever. WHAT GOOD ARE FORCES??? Forces and mass tell us the acceleartion of a mass...but we already know all we need to know about acceleration...they are the SAME! So the forces and masses involved are of no consequence to the claim. Please make sure this is clear to you before you continue. It is of fundamental importance)
Alright. <b>So, to be clear, the proposition and it's corollary are all I claim.</b> Here are the details of why Prop. 1 is TRUE.
For every time t>=0 there will be some unique acceleration given by some vector function a:R->R^3
Acceleration is by deffinition: X''(t)
So we have the 2nd order differential equation X''(t) = a(t), for each object
X_p1''(t) = a(t)
X_p2''(t) = a(t)
In physics, if we want to describe the motion of a a mass, we want to know X(t) for times, t>=0.
From X(t), although interesting on its own right, we can find the other interesting stuff...velocity, acceleration, jerk, kinetic energy, whatever. Of course, some of these things we might have known before hand, but you get the picture.
So Lets solve the equations.
Integrating once we get the velocity funtions:
X_p1'(t) = X'(0)+S[0,t]a(t*)dt*
X_p2'(t) = X'(0)+S[0,t]a(t*)dt*
integrating one last time we get we get the position functions:
X_p1(t) = X_p1(0)+X'(0)t+S[0,t](S[0,t*]a(t**)dt**)dt* (t* and t** are just dummy variables, of course)
X_p2(t) = X_p2(0)+X'(0)t+S[0,t](S[0,t*]a(t**)dt**)dt*
(note, dont worry about the scary looking S[0,t](S[0,t*]a(t**)dt**)dt*, its just some function of time that X_p2 and X_p1 have in common)
So we see, given the initial conditions, there is a unique solution. To formally establish the uniqueness and existence of solutions to classes of ordinary differetial equations, such as these, see Picards's existence theorem.
http://mathworld.wolfram.com/PicardsExistenceTheorem.html
So is the motion of the two the same? YUP. Inspect above...we see:
X_p1'(t) = X_p2'(t), the velocity graphs are the same.
X_p2(t) = (X_p2(0)-X_P1(0))+X_p1(t), welp...X_p2(0)-X_p1(0) is a constant, it's just the two object's COM initial displacement from each other. So Yes, Their position graphs only differ by a uniform translation.
So, it's been shown they have the same motion, based on the assumptions I stated. Hence they will also have the same turning radius regardless of mass.
QED
It could be made more rigerous...but there isn't much point. I assume there are no anal mathematician's here that want me to tell you why I can take the integral of certain functions and such, right?

If there are any questions about any of how/why I defined some things in here, feel free to ask

I, again, appologize for the horrible ASCII math.
Just curious, Mr Mudd, where are the flaws in this argument? Even if there were minor ones, you agree with the proposition and collorary's statements, right?
Edited by - feynman on Aug 14 2004 6:45 PM