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brachistochrone    
最短时程; 最速降线; 捷线

最短时程; 最速降线; 捷线


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  • Is there an intuitive reason the brachistochrone and the tautochrone . . .
    The brachistochrone problem asks what shape a hill should be so a ball slides down in the least time The tautochrone problem asks what shape yields an oscillation frequency that is independent of amplitude
  • Brachistochrone Problem w Initial Velocity - Physics Forums
    The discussion revolves around the Brachistochrone problem, specifically addressing the scenario where an object has a non-zero initial velocity Participants explore the implications of this condition on the optimal path between two horizontal points, the calculations involved, and the potential existence of formulas for such cases
  • Brachistochrone. Why a curve at all? - Physics Forums
    The discussion centers around the brachistochrone problem, specifically questioning why the solution involves a curve rather than a straight line when moving from a higher point to a lower point under gravity
  • energy conservation - Brachistochrone problem with initial velocity . . .
    The Brachistochrone problem is usually presented with the having a ball dropped into the slide with initially zero velocity and at position $(x, y)=(0, 0)$ I would like to know the more general so
  • Why do I need the Beltrami identity to solve the brachistochrone problem?
    It seems to be the same answer, just expressed very differently and very much more difficult to get the answer from the method avoiding Beltrami identity Why should textbook authors waste paper talking about solution methods that are unnecessarily difficult?
  • How can you deepen your understanding of the brachistochrone problem . . .
    The discussion centers on the brachistochrone problem, a classic physics and calculus of variations challenge The original poster has explored the cycloid's parametric equations using Bernoulli's method and created programs to visualize the curve However, they seek further guidance on advancing their understanding, particularly in areas such as geometric curves, friction in physics, and
  • Why Does a Cycloid Curve Minimize Travel Time? - Physics Forums
    The discussion centers on the brachistochrone problem, specifically exploring why a cycloid curve minimizes travel time for a ball rolling down a ramp Participants seek a semi-detailed understanding of the physics involved, touching on both conceptual and mathematical aspects of the problem
  • Solving the Brachistochrone problem with friction - Physics Forums
    The discussion focuses on solving the Brachistochrone problem with friction, specifically analyzing the differential equation derived from the Euler-Lagrange equation The key point is understanding how to derive the parametric forms for x and y in terms of theta, as outlined in steps (32) and (33) of the Wolfram Alpha page The user seeks clarification on whether setting dy dx = cot (theta 2
  • How Do You Calculate Time in the Brachistochrone Problem?
    The Brachistochrone problem involves determining the curve along which a ball travels between two points in the least amount of time The curve is defined as a cycloid, represented in parametric form by the equations x (θ) = C (θ - sin (θ)) and y (θ) = -C (1 - cos (θ)) To calculate the travel time, the integral t_ {12} = ∫_ {T_1}^ {T_2} (√ (1 + (y')²) √ (2gy)) dx must be





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