A mechanics question with "constant acceleration"

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Main problem is with part (b). I got it right up to the accelerations with -11.6 and 4.4 respectively.

Then I dont really see the reason why we have to break it up to two parts as given in the solution. We should be able just to use the constant acceleration formulae and just to find the time where the displacement is the SAME (time just before collison). Even if Q stops moving up, it would change direction then going downwards...(That's the difference between mine answer and the "correct" answer?)

What am I thinking why?

Model answer given below.

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