M1 tension string question

Is it because the question is asking for distance travelled and you have calculated maximum displacement during the ballistic section of the motion.

I think you have worked out the distance that A travels from the moment B hits the plane and when A is at its maximum height. If you have, you have to double the distance because the point at which the string becomes taut again will be the position where A was when B hit the plane.
Please post your workings though if this doesn't make sense.

I did exactly what you said which is working out the Max distance that a can reach. Do I have to take into account of the taut string in the question? I think I have misunderstood it. thanks.

You have correctly worked out the max height of A. To get the 0.6m answer, you need to consider when the string will become taut again.
The string will be taut again when particle A comes back down to the same height that it was at when B hit the plane. So A will move 0.3m up, and then 0.3m down, back to the same position, and there will now be tension in the string - string is taut.

They are both moving at the same time, but when there is tension in the string (the string is taut) the acceleration is the one that you calculated in the earlier part of the question. As soon as B hits the plane, the string is no longer taut and the only force now acting on A is weight - so it's acceleration is g - it's in free fall.
The question states that we assume B stays on the ground and does not get lifted at all.

But why does a need to move back down then?