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Question Number 4140 by prakash jain last updated on 29/Dec/15
Four persons a, b, c, d are standing at four  vertices of square ABCD.  All four start moving simultaneously such  a is always moving towards b on a straight  line between a and b. Similary b is always  moving directly towards c, c is directly  moving towards d and d is directly moving  towards a.  Do they all meet? How much will be the  distance travelled by each when they meet?
$$\mathrm{Four}\:\mathrm{persons}\:\mathrm{a},\:\mathrm{b},\:\mathrm{c},\:\mathrm{d}\:\mathrm{are}\:\mathrm{standing}\:\mathrm{at}\:\mathrm{four} \\ $$$$\mathrm{vertices}\:\mathrm{of}\:\mathrm{square}\:\mathrm{ABCD}. \\ $$$$\mathrm{All}\:\mathrm{four}\:\mathrm{start}\:\mathrm{moving}\:\mathrm{simultaneously}\:\mathrm{such} \\ $$$$\boldsymbol{\mathrm{a}}\:\mathrm{is}\:\mathrm{always}\:\mathrm{moving}\:\mathrm{towards}\:\boldsymbol{\mathrm{b}}\:\mathrm{on}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{between}\:\boldsymbol{\mathrm{a}}\:\mathrm{and}\:\boldsymbol{\mathrm{b}}.\:\mathrm{Similary}\:\boldsymbol{\mathrm{b}}\:\mathrm{is}\:\mathrm{always} \\ $$$$\mathrm{moving}\:\mathrm{directly}\:\mathrm{towards}\:\boldsymbol{\mathrm{c}},\:\boldsymbol{\mathrm{c}}\:\mathrm{is}\:\mathrm{directly} \\ $$$$\mathrm{moving}\:\mathrm{towards}\:\boldsymbol{\mathrm{d}}\:\mathrm{and}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{directly}\:\mathrm{moving} \\ $$$$\mathrm{towards}\:\boldsymbol{\mathrm{a}}. \\ $$$$\mathrm{Do}\:\mathrm{they}\:\mathrm{all}\:\mathrm{meet}?\:\mathrm{How}\:\mathrm{much}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the} \\ $$$$\mathrm{distance}\:\mathrm{travelled}\:\mathrm{by}\:\mathrm{each}\:\mathrm{when}\:\mathrm{they}\:\mathrm{meet}? \\ $$
Commented by prakash jain last updated on 30/Dec/15
yes.
$${yes}. \\ $$
Commented by Rasheed Soomro last updated on 30/Dec/15
If all are moving contineously, their direction  of movement should change at every point.  That means they are moving along curved  path.     Their movements for short time interval δt   can be cosidered in straight line. If I have  understood....
$$\mathrm{If}\:\mathrm{all}\:\mathrm{are}\:\mathrm{moving}\:\mathrm{contineously},\:\mathrm{their}\:\mathrm{direction} \\ $$$$\mathrm{of}\:\mathrm{movement}\:\mathrm{should}\:\mathrm{change}\:\mathrm{at}\:\mathrm{every}\:\mathrm{point}. \\ $$$$\mathrm{That}\:\mathrm{means}\:\mathrm{they}\:\mathrm{are}\:\mathrm{moving}\:\mathrm{along}\:\mathrm{curved} \\ $$$$\mathrm{path}. \\ $$$$\:\:\:\mathrm{Their}\:\mathrm{movements}\:\mathrm{for}\:\mathrm{short}\:\mathrm{time}\:\mathrm{interval}\:\delta\mathrm{t}\: \\ $$$$\mathrm{can}\:\mathrm{be}\:\mathrm{cosidered}\:\mathrm{in}\:\mathrm{straight}\:\mathrm{line}.\:\mathrm{If}\:\mathrm{I}\:\mathrm{have} \\ $$$$\mathrm{understood}…. \\ $$
Commented by Rasheed Soomro last updated on 31/Dec/15
•They will meet at the center of the square.  •When they will meet, they will be at the  distance equal to half of the diagonal of the   square, from their original positions.
$$\bullet\boldsymbol{\mathrm{T}}\mathrm{hey}\:\mathrm{will}\:\mathrm{meet}\:\mathrm{at}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}. \\ $$$$\bullet\mathrm{When}\:\mathrm{they}\:\mathrm{will}\:\mathrm{meet},\:\mathrm{they}\:\mathrm{will}\:\mathrm{be}\:\mathrm{at}\:\mathrm{the} \\ $$$$\mathrm{distance}\:\mathrm{equal}\:\mathrm{to}\:\mathrm{half}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diagonal}\:\mathrm{of}\:\mathrm{the} \\ $$$$\:\mathrm{square},\:\mathrm{from}\:\mathrm{their}\:\mathrm{original}\:\mathrm{positions}. \\ $$

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