Path that requires the least time to travel along

Click For Summary

Homework Help Overview

The discussion revolves around a problem in physics related to determining the path that requires the least time to travel along, likely involving concepts from kinematics and calculus. Participants are exploring the relationship between trajectory and time of travel in a circular context.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the initial setup involving a circle and the implications of connecting points with straight lines. There is an exploration of the trajectory and its justification, as well as considerations of gravitational acceleration and its components along the path.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts and sketches. Some guidance has been offered regarding the trajectory and potential simplifications, but no consensus has been reached on the best approach or solution.

Contextual Notes

Participants are working within the constraints of a homework problem, which may limit the information available for discussion. There is an emphasis on justifying reasoning and exploring different angles of approach.

bbal
Messages
5
Reaction score
0
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
 
Physics news on Phys.org
Show your work so far.
 
  • Like
Likes   Reactions: Frigus
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...
 
  • Like
Likes   Reactions: bbal

Similar threads

Electric Circuit: Path of Least Resistance
  • · Replies 1 ·
Replies
1
Views
1K
Which Path is Quickest? Investigating Ball Bearings on an Inclined Tube
  • · Replies 14 ·
Replies
14
Views
3K
Shortest distance along the shore and into the lake
  • · Replies 6 ·
Replies
6
Views
1K
Calculating Time Dilation for Space Travelers
  • · Replies 4 ·
Replies
4
Views
1K
Closed path term in Thermodynamics
  • · Replies 38 ·
2
Replies
38
Views
4K
How Does Changing Distance Affect Microwave Path Difference?
  • · Replies 7 ·
Replies
7
Views
1K
Vehicle moving along a path tangent to the Earth surface
  • · Replies 1 ·
Replies
1
Views
881
Four Snails Traveling on a Plane -- Do they cross paths?
  • · Replies 33 ·
2
Replies
33
Views
3K
How to find eddy current (circular) path in this pendulum experiment?
  • · Replies 9 ·
Replies
9
Views
1K
Undergrad Def Proper Time GR: Half or Integral Along Path?
  • · Replies 17 ·
Replies
17
Views
3K