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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

A research team of students in the Physics department of University of South Carolina has been watching the movement of a flea for 60 minutes without interruption.

Each student continuously watched the movement for exactly one minute and during this time, each of them saw the flea move by 1 meter during his 1-minute observation. What is the minimum and maximum distance which the flea could have moved during the 60 minutes of observation? (distance, not displacement)

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

In what directions is he walking? Any direction? How many dimensions, 1,2, 3... is he moving in?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

In any direction, but we are not really interested in the direction, only the absolute distance he has "traveled".

bobbym wrote:

Hi;

In what directions is he walking? Any direction? How many dimensions, 1,2, 3... is he moving in?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

On a plane or in 3D?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

Let's assume he (or she??) makes jumps of 1m each. On a plane.

bobbym wrote:

On a plane or in 3D?

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

I do not then understand the question. Distance is a scalar, it means the amount of ground covered. Therefore if the flea jumped 1m each minute, every minute, the maximum and minimum distance is the same, 60.

Please clarify the problem further. Are you thinking of a 2D random walk?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

I don't know exactly but I guess it has to do with the series of "jumps" the flea is doing. That is, if he starts a second jump after the completion of the first, but does not "land" within the minute of observation, but in the next one, and then he jumps again... I don't know - just guessing!!

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hmmm, do you have the exact question?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

This is how a friend gave it to me. I think it was originally posted in the Kvant magazine.

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

I have searched for the question and Kvant magazine, I did not find anything.

I think the question as posed is incomplete and I am unable to say more about the flea than I did in post #6.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 1,086

The catch lies in the wordings. The students have observed the distance moved by flee, not the displacement. Therefore the maximum distance moved is 1 meter and minimum distance moved is also 1 meter. The other interpretation is the minimum distance it has moved is 0. That is if you make a tabulation of distance vs time(in seconds) during this interval the disatnce is 0 at time 0 and goes on increasing to become 1 meter at 60 seconds, the minimum being 0 and maximum being 1 meter.

**{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha{Gods rejoice at those places where ladies are respected.}**

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**Relentless****Member**- Registered: 2015-12-15
- Posts: 624

Hi thickhead, I think you missed that the flea was observed for 60 minutes, not 60 seconds. Besides that, you and I agree with bobbym, and I like the tricky second interpretation you mention

I think there must be another interpretation, however. The distinction between distance and displacement would only be useful if the flea moved in multiple directions. So the 1-metre movement observed by each student must refer to displacement, while the question refers to distance. The problem seems a bit silly then though... mustn't the maximum then be merely the maximum speed per hour the flea can achieve whilst changing direction rapidly?

*Last edited by Relentless (2016-05-07 00:31:19)*

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**thickhead****Member**- Registered: 2016-04-16
- Posts: 1,086

Each had observed the distance ( not displacement) for 1 minute. for 60 minutes it is the sum of distance covered 1 minute each totalling 60 meters.

**{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha{Gods rejoice at those places where ladies are respected.}**

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**Relentless****Member**- Registered: 2015-12-15
- Posts: 624

Agreed. Or... it refers to displacement and the minimum is 60 metres while the maximum is the highest speed the flea can achieve while changing direction.

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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

Guys,

Although I have absolutely no clue on how to solve it, I suspect the problem refers to 1-meter "jumps" that the flea does, so maybe during the 1-minute observation by each student, one jump may have "started" (so the flea jumped upwards) but maybe he "landed" at the next student's minute - if you see what I mean. So this 1-meter was not accounted, neither to the previous student's observation, nor to the next one's, because the first one did not see the flea "landing", while the other one did not see him "taking-off" I will leave the rest to you

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**iasonidis****Member**- Registered: 2017-02-11
- Posts: 2

Hi all,

I think I've seen this somewhere else, with a slightly different wording. I believe it says that the insect moves by jumps on a straight line and to the same direction.

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**chen.aavaz****Member**- Registered: 2016-05-05
- Posts: 34

I think I am beginning to understand:

See image here

First of all, the flea moves in jumps, on a straight line and to the same direction (I am just thinking out loud...).

Each student observes the flea for exactly one minute and during this minute, it has made, say, n jumps of x centimeters each, with total length n*x=1m.

The last jump has not yet settled within the observation minute, and so it has not been accounted during the 1st student's observation. Similarly, since the second student only sees the flea once the previous jump settles down, this jump is not accounted to his minute either. So again, he observes n jumps of x centimeters each (assuming that the jumps are of equal length). Yes, I know, too many assumptions, but I am running out of ideas!!

So now the question lies to determine the maximums and minimums.

Clearly n>=1

Also x <= 1m because if x>1m then each jump would never settle within the observation minute, so the students would not see the flea to move by 1m.

As we see in my draft drawing, the small portion of the "unfinished" jump (which is obviously <x) gradually moves towards the right and then there is another unfinished jump which begins right before the end of the 2nd minute.

...I don't know how to continue, unless we make some more assumptions, like, for example, that the minimum x is, say, 1 centimeter?

Any help, anyone?

*Last edited by chen.aavaz (2017-02-20 08:39:51)*

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**salem_ohio****Member**- Registered: 2016-03-11
- Posts: 37

If n=1 then we have 1 complete jump in each minute plus some unfinished jumps before and after. Each jump must be exactly 1m, right? Then the MINIMUM is 60m? Not sure.

Of course the maximum is a lot more difficult...

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