Relationship between linear and circular velocity

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Discussion Overview

The discussion revolves around the relationship between linear and circular velocity, specifically focusing on the time it takes for a runner to complete a circular arc compared to a straight path, assuming constant linear velocity. Participants explore the implications of centripetal force and acceleration on running speed and time taken for different paths.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant suggests that if distances and speeds are the same, then the time taken must also be the same, referencing the formula distance = speed * time.
  • Another participant questions the clarity of the original question, emphasizing that a runner covers the same distance regardless of the path taken if speed is constant.
  • A later reply introduces the concept of centripetal force, suggesting that it may affect the runner's forward momentum and thus the time taken to navigate a circular path compared to a straight one.
  • One participant proposes that the maximum speed on a curve would be less than on a straight path due to the need for acceleration, indicating a possible relationship with angular acceleration.
  • Another participant states that centripetal acceleration does not affect forward velocity if the runner maintains maximum speed in both scenarios.
  • One contribution mentions that power remains constant, but additional force is required to overcome centrifugal force on a circular path, which could impact speed.
  • Another participant asserts that kinematics can demonstrate that the times are equal, suggesting a straightforward approach to the problem.
  • One participant notes that if the curve isn't too steep, the runner does not expend significant energy to create centripetal acceleration.

Areas of Agreement / Disagreement

Participants express differing views on whether the time taken to run a circular path is the same as that for a straight path, with some asserting equality based on kinematics while others introduce factors like centripetal force and acceleration that could complicate the relationship. The discussion remains unresolved regarding the impact of these factors on time taken.

Contextual Notes

There are limitations in the assumptions made regarding the runner's speed and the effects of centripetal force, as well as the conditions under which the comparisons are made. The discussion does not resolve the mathematical implications of these factors.

Robert Vaughan
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My Math and Physics knowledge has eroded from over 30 years of non-use though I can still integrate e^x with the best of you!

My son wants to know how to determine the time it takes to travel/complete a circular arc in terms of the time it takes to travel the same distance going in a straight line ... assuming in both cases that the "linear velocity" is the same and is constant.

I may be expressing it incorrectly but I think he is trying to determine the additional time it would take a runner to travel the same distance on a circular track ... relative to the time it would take the same runner to travel that distance on a straight track ... both with identical surfaces.

Thanks!
 
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Robert Vaughan said:
My Math and Physics knowledge has eroded from over 30 years of non-use though I can still integrate e^x with the best of you!

My son wants to know how to determine the time it takes to travel/complete a circular arc in terms of the time it takes to travel the same distance going in a straight line ... assuming in both cases that the "linear velocity" is the same and is constant.

I may be expressing it incorrectly but I think he is trying to determine the additional time it would take a runner to travel the same distance on a circular track ... relative to the time it would take the same runner to travel that distance on a straight track ... both with identical surfaces.

Thanks!
If the distances are the same and the speeds are the same, then the time must also be the same. Distance= speed * time, regardless of if it is linear or not. Maybe that isn't what he was asking...?
 
Not sure I'm getting the point of the question. A runner will cover the same distance regardless of his path, as long as his speed is constant.
 
Sorry for the confusion ... my terms were obviously incorrect ... in both cases, the runner is running at his "maximum speed" so I suppose you could say the expended energy is the same ... his understanding is that when running on a circular path, centrifical force would come into play and his "forward momentum" would be reduced ... so that it would take longer to navigate the arc than it would to travel the same distance on a straight course.

Again, the terms may be incorrect but I hope the intent of the question is clearer.
 
I guess his maximum speed on a curve would be less than his maximum speed on a straight because the runner would still be accelerating on the curve. The force required to do this acceleration would be his mass times the angular acceleration which (correct me if I'm wrong!) is v2/2 (you might want to check that cos I don't really use this very often).

Actually i know that's wrong cos you need the radius in there somewhere!
 
its mv^2/r

it would take the same amount of time assuming the guy is running at his maximum speed in both cases, centripetal acceleration does not effect forward velocity.unless your question was more along the lines of how much longer will it take to run around a circle as compared to running across the circle.
 
The way I'm thinking about it you say the power is constant.

Power = work/time = force*speed = constant

because on the circle the runner needs more force to overcome the centrifugal force, his speed must drop. The additional force he will need is mv^2/r.
 
As the first answers to this post mention, it can be shown with simple kinematics that the times are equal. No need to think about anything else than kinematics.
 
As long as the curve isn't too steep, the runner is not expending a significant amount to create centripedal acceleration.
 

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