Galileo’s Swiftest Descent Problem Revisited: Complete Analytic Solution

Selg, Matti

doi:10.1080/00029890.2022.2142031

Cited by 2 publications

(2 citation statements)

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“…Let 0, 1, 2 and 3 be the points on a circle, such that 0  j 0 < j 1 < j 2  j 3  p. Suppose a mass point at rest starts to fall from point 3, moving along an arbitrary path to point 2. The descent time along the chord from point 2 to point 1 can then be calculated as follows [6] (see also [3], Appendix B):…”

Section: A Few Preliminary Analytical Resultsmentioning

confidence: 99%

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Generalized approach to Galileo’s swiftest descent problem on a circle

Selg¹

2023

PEAS

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Nearly 400 years ago Galileo Galilei posed a conjecture about the fastest descent on the lower half of a circle. Namely, he assumed that the descent along the circular arc itself is faster than along any broken line of chords. Galileo studied the case when the initial speed is zero, and the paths end in the lowest point of the circle, but he made a guess that the final conclusion remains the same if the particle initially at rest falls to a starting point where the speed is not zero. Galileo was right, as is now rigorously proved. However, this intuitive idea cannot be extended to a starting point on the upper half of a circle. Indeed, as we demonstrate, the fastest descent would then correspond to a finite (not infinite!) number of connected chords. Moreover, we present an extrapolation method which can be applied to determine the optimal number and the positions of all these chords for any starting point on a circle.

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Section: A Few Preliminary Analytical Resultsmentioning

confidence: 99%

“…Indeed, as Galileo obviously was aware of, and as has been pointed out by several authors [4,5], Proposition 36 cannot be extended to a situation where the particle is not initially at rest. A rigorous analytical proof for this case has been given only very recently [6].…”

Section: Introductionmentioning

confidence: 99%

Generalized approach to Galileo’s swiftest descent problem on a circle

Selg¹

2023

PEAS

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An energy perspective on Galileo’s kinematic paradox

Hailu

2024

Eur. J. Phys.

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The connection between kinematics and work-energy principles is often not emphasized enough for students to appreciate fully. Highlighting the relationship between the kinematic equations and the conservation of total mechanical energy is crucial. We revisit the solution to Galileo's kinematic paradox from an energy perspective. Galileo stated that the time it takes for an object to travel from the top to the bottom point of a vertical circle is the same regardless of its path. While mathematical proofs of this statement using kinematic approaches are well-documented, this paper explores the problem from an energy standpoint, using experimental evidence to elucidate the relationship between kinematics and energy.

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Galileo’s Swiftest Descent Problem Revisited: Complete Analytic Solution

Cited by 2 publications

References 3 publications

Generalized approach to Galileo’s swiftest descent problem on a circle

Generalized approach to Galileo’s swiftest descent problem on a circle

An energy perspective on Galileo’s kinematic paradox

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