"When you both stop, I stop" - Centre Of Mass

Two particles \(A\) and \(B\) are situated on a horizontal table top of infinite surface area whose surface is considered as the \(x-y\) plane. The mass of \(A\) is \(1~kg\) and that of \(B\) is \(2~kg\). At time \(t=0\), particle \(A\) is at \((3,0)\) and particle \(B\) is at \((0,9)\). Both particles are given initial velocites:

  • \(3~ms^{-1}\) to \(A\) along positive \(x\)-axis and

  • \(6~ms^{-1}\) to \(B\) along positive \(y\)-axis.

If the coefficient of friction between each particle and the table top is \(\mu = 0.2\) & the coordinates of the position of the centre of mass of \(A\) and \(B\), where it comes to rest can be expressed as \(\left(\dfrac{a}{b},\dfrac{c}{d}\right)\), where \(\left\{a,b\right\}\) and \(\left\{c,d\right\}\) are pairwise co-prime positive integers, then enter your answer as

\(\displaystyle (c+d)-(a+b)\).

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