Path that requires the least time to travel along

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SUMMARY

The discussion focuses on optimizing travel time along a circular path by analyzing gravitational acceleration and trajectory angles. Participants explore the concept of connecting the top of a circle to various points along its circumference, emphasizing the need to minimize travel time through mathematical justification. The approach involves calculating the gravitational component parallel to the rail and varying the angle α to find the optimal trajectory. The conversation suggests leveraging circle theorems for a more straightforward solution.

PREREQUISITES
  • Understanding of basic physics concepts, specifically gravitational acceleration.
  • Familiarity with circle geometry and theorems.
  • Knowledge of calculus, particularly in relation to optimization problems.
  • Ability to sketch and analyze trajectories in a circular motion context.
NEXT STEPS
  • Study the principles of gravitational acceleration in circular motion.
  • Learn about circle theorems and their applications in trajectory optimization.
  • Explore calculus techniques for minimizing functions, particularly in physics contexts.
  • Investigate practical applications of trajectory optimization in engineering and physics.
USEFUL FOR

Students of physics, mathematicians, and engineers interested in optimizing motion along circular paths and understanding the interplay between geometry and gravitational forces.

bbal
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Homework Statement
We have slope, over which there's a point (P). The point is connected to the slope with a straight line. Find the line, a small ball would travel along the fastest.
Relevant Equations
S=(at^2)/2
IMG_20201102_194520.jpg
 
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Show your work so far.
 
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DaveC426913 said:
Show your work so far.
It's basically all you can see on the picture. I took it as a starting, that if we connect the "top" of a circle to any other point of the circle with a straight line, the time to travel along each would be the same. Then I tried to sketch a circle, such that, point P is on "top" of it and the slope is tangent to it.
 
Well funnily enough you actually drew the correct trajectory on your left hand diagram, but can you justify it?

The obvious, but not particularly elegant, approach, is to consider an arbitrary trajectory at some angle ##\alpha## to the downward vertical through the point P, find the component of gravitational acceleration parallel to the rail, find the length of this rail, and vary ##\alpha## in such a way to minimise the time.

The answer might suggest a simpler line of reasoning! If you remember your circle theorems...
 
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