# Relationship between linear and circular velocity

## Main Question or Discussion Point

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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russ_watters
Mentor
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...?

Doc Al
Mentor
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 wanna 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.