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Great! This is a completely acceptable alternate specification of
the problem which will yield the same fundamental physics.
Correct to here.
v = F/m t [7]
No. I get
v = F/2m t [7_corr]
We are asked to calculate the CM position (x1+x2)/2
which by equation [4] is just
X = x1 + L/8. [8]
No. I get
X = x1 - L/8 [8_corr]
At any rate, using the corrected equations [7_corr] and [8_corr],
I get
X = -L/8
+ m/F v^2
+ 1/2 integral (x1' - x2') dt
... we can easily perform the integral and find that
1/2 integral (x1' - x2') dt = 1/2 (x1 - x2)
= 1/2 (L/4)
= L/8
I hope my explanation has been satisfactory.