Is it possible to solve for “t?”

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  • #51
pbuk
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hmm... i thought there was no exact solution for t when A and B are both rational... (or t is irrational)
It's got nothing to do with being irrational, it is simply not helpful to talk about this task as 'endless'.

It is true that it is not possible to write down an 'exact' solution to the problem, but neither is it possible to write down the 'exact' value of 1/3 (a rational number) in the decimal number system, would you refer to that as endless?

There are things in mathematics which are 'endless', such as searching for the limit of the sum ## 1 + \frac12 + \frac13 + \frac 14 ... ##, but that is not the situation we have here.
 
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  • #52
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It is true that it is not possible to write down an 'exact' solution to the problem, but neither is it possible to write down the 'exact' value of 1/3 (a rational number) in the decimal number system, would you refer to that as endless?
Excellent point that seems to get to the heart of what the OP is asking about.
I can solve the equation ##3x = 1## algebraically to get the exact answer, but it will take an "endless" number of digits to write the exact value as a decimal fraction.

But so what? Rational or irrational makes just about zero difference here.

My suspicion is that what the OP is trying to get at is that the object can slide down the tube in a finite amount of time, but writing all of the decimal digits would take an infinite amount of time. I could be wrong, but if not, this is not something to be concerned about.
 
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  • #53
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I added an extra set of parenthesis:

A=−(B/(2((pi−t)+sin(t))))(1−cos(t))

When B and t are 1, A should be -0.077...
Just to confirm, would it be accurate to make the following statement—

“Whenever t & B are rational, A is irrational. Whenever A & B are rational, t is irrational.”

?
I suppose I thought that, when A & B are rational (each can be expressed as a ratio between integers) that, since the formula can’t be rearranged algebraically to solve for t, that t might not be able to be expressed as a ratio between integers and hence t would be “irrational” though I’m not sure.

I was imagining, if t is irrational, it leading to a scenario where someone with a fast computer solves t to very many digits, and from t and B calculates r, and from r, g and t, calculates the travel time and says “this is the quickest route.” But then someone else calculates to more digits, and finds an even faster time. Could this lead to a scenario where one can’t find the “quickest” route because someone with a faster computer and more time can always calculate t to more digits than the person before (more digits of t affecting and altering the proposed “quickest” cycloid generating radius), so as time goes on ever faster routes can be found ad nauseum?
 
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I was imagining, if t is irrational, it leading to a scenario where someone with a fast computer solves t to very many digits, and from t and B calculates r, and from r, g and t, calculates the travel time and says “this is the quickest route.” But then someone else calculates to more digits, and finds an even faster time. Could this lead to a scenario where one can’t find the “quickest” route because someone with a faster computer and more time can always calculate t to more digits than the person before (more digits of t affecting and altering the proposed “quickest” cycloid generating radius), so as time goes on ever faster routes can be found ad nauseum?
So, two points:

(1) I think that what others are trying to say is that even if the "exact solution" for t is a number with infinitely many digits, the problem you seem to be worried about holds for any such number. For example, maybe I know the answer is 1/3 (a rational number). If I have to write that out as a decimal, I know exactly how to do that - start with 0.33 and keep adding 3's on the right. So it would take me an infinite amount of time to write it down exactly, but not nearly an infinite amount of time to write it out to a reasonable degree of precision.

Apply the same reasoning to a number like ##\pi## (an irrational number). If you give me a radius, I can calculate the circumference ##2\pi r## of a circle to arbitrarily many digits, depending on how many digits of ##\pi## I want to compute and use. No, it will never have infinitely many digits, but there's never going to be an application in which you need to know infinitely many digits. Knowing the value of the circumference that's more precise than any of your measuring devices, for example, is more than enough. More digits than that become redundant unless you're going to build a better tape measure.

(2) Both in the circle example and in your example of an extremal trajectory, using more digits will not radically change your final answer. Rather, you should expect that if you keep computing with more and more precision, your circumference or trajectory will be converging to some particular thing. No, you won't be able to say what it is with infinitely many digits, but you'll have, for example, a trajectory that changes so little as you introduce more digits for t that you very quickly stop being able to measure the resulting differences in travel time. Think of computing with t to 10 digits, and then with 15, and getting two trajectories. They will probably be similar to the point that a stopwatch could not measure the difference in time between taking each path, and indeed they're probably similar enough that no instrument could chart a course that knows how to distinguish between them. Furthermore, a computer would take very little time to do either of these calculations.

In fact, you'll find that you waste much more time trying to calculate to infinite digits than to just travel on the trajectory that's good to 10 digits, even while your measurable time spent on each of these paths would not be different.

Am I understanding correctly that you think there's some paradox in taking infinite time to calculate a trajectory that minimizes time? I'm not sure whether you just find this kind of ironic or whether it really bothers you mathematically. Hope this helps!
 
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  • #55
pbuk
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I suppose I thought that, when A & B are rational (each can be expressed as a ratio between integers) that, since the formula can’t be rearranged algebraically to solve for t, that t might not be able to be expressed as a ratio between integers and hence t would be “irrational” though I’m not sure.
Nearly all numbers are irrational but they don't seem to worry about it and nor should you :biggrin:

I was imagining, if t is irrational, it leading to a scenario where someone with a fast computer solves t to very many digits, and from t and B calculates r, and from r, g and t, calculates the travel time and says “this is the quickest route.” But then someone else calculates to more digits, and finds an even faster time. Could this lead to a scenario where one can’t find the “quickest” route because someone with a faster computer and more time can always calculate t to more digits than the person before (more digits of t affecting and altering the proposed “quickest” cycloid generating radius), so as time goes on ever faster routes can be found ad nauseum?
No, there is only one fastest route.

Bear in mind that you have used the approximate value of 9.8 m/s2 for g: there is no point specifying your solution to more than 3 significant figures because it will change if use use a different value of g (note that g varies by about 0.05 m/s2 between the equator and the poles and there are also local variations due to altitude and geophysical anomalies).
 
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  • #56
WWGD
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I assume the Implicit Funcion Theorem could give a definitive answer on the possibility of isolating t?
 
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  • #57
pbuk
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Not my area of expertise but won't the Implicit Function Theorem just give us existence not construction? We don't have a problem with existence - we don't even have a problem with calculation; this is simple root finding for a (in the region of interest) monotonic function so we can achieve arbitrary precision in logarithmic time.
 
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  • #58
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I assume the Implicit Funcion Theorem could give a definitive answer on the possibility of isolating t?
I don't think so. I agree with @pbuk that this theorem just talks about the existence of a function. From the wiki page, https://en.wikipedia.org/wiki/Implicit_function_theorem
The implicit function theorem gives a sufficient condition to ensure that there is such a function.
Not my area of expertise but won't the Implicit Function Theorem just give us existence not construction?
That's how it seems to me.
 
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  • #59
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I don't think so. I agree with @pbuk that this theorem just talks about the existence of a function. From the wiki page, https://en.wikipedia.org/wiki/Implicit_function_theorem

That's how it seems to me.
Agreed, but maybe a 'no', not possible to represent even locally, would shed some light. Let me look at the theorem again see if I can find a constructive corollary of some sort.
 
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  • #60
pbuk
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It is probably worth a summary as there have been a few diversions and dead ends on the way.

We are trying to find the shortest journey time between points ## A ## and ## B ## at the same elevation for an object under the influence of gravity constrained without friction to a path. The object has initial velocity of ## v_0 ## in the initial direction of travel, and arrives at B with equal speed.

By other work referenced, we know that the solution is part of a cycloid defined by
$$ \begin{align*} x &= r (t - \sin t) \tag 1 \\
y &= r (1 - \cos t) \tag 2 \end{align*} $$
at point ## A ## we must have
$$ mgy_A = 0.5mv_0^2 \implies y_A = \frac{v_0^2}{2g} \tag 3 $$
and we know the distance ## X ## between ## A ## and ## B ## and that ## x_A = x_B ## so
$$ 2 x_A + X = 2 \pi r \implies X = 2 (\pi r - x_A) \tag 4 $$

We now have two unknowns, ## t_0 ## (the value of the parameter t at X) and ## r ## (the generating radius). There are a number of ways of proceding: the OP (if I remember rightly) has estimated ## t_0 ## and used (2) and (3) to find the corresponding ## r ##; substituted this into (1) and (4) to find the resulting ## X ## and reduced the error by bisection.
 
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  • #61
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2 comments...
By other work referenced, we know that the solution is part of a cycloid defined by
I believe that equations 1 & 2 specify a cycloid that proceeds above the x axis from the origin, but the needed path is an upside-down or inverted cycloid that proceeds below the x axis from the origin.


we know the distance X between A and B and that xA=xB so
I believe you meant yA=yB instead of xA=xB
 
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  • #63
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and we know the distance X between A and B and that xA=xB so
(4)2xA+X=2πr⟹X=2(πr−xA)

We now have two unknowns, t0 (the value of the parameter t at X)
There may be an error in equation 4... I think the equation is true for a complete cycloid when you know the horizontal distance between the cusps (except 2xA should simply be xA), but isn’t true when the cycloid is truncated by a certain distance in the y axis when you know the horizontal distance between the truncated cusps.

I believe xA + X = 2pi * r only when the cycloid hasn’t been trimmed in the y axis by specifying a range for t other than 0<=t<=2pi

M6le1.gif
 
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