Momentum

The inextensible string becomes taut. They must be moving with the same velocity.

How do you know?? I don't understand this part

How fast does a caravan move when coupled to a car?

Same speed as the car. So applying this analogy to the question, the particles must move with same speed..??

But then, what is the point of giving the length of the string and the initial separation of the particles?

So initially they have different speeds (one is zero). As the string becomes taut, you have to apply conservation of momentum to calculate the new speed(s)

Thanks, I understand this part.

I think it makes sense to me now that the speeds must be the same. I think the separation of particles and length of string are not needed to work out the answer

Thanks for your help!

The only thing it affects is that the string is initially slack then becomes taut.

I have some sympathy with the OP here. The given answer is clearly based on the assumption that P and Q will travel with the same speed once the string has initially become taut, but it's not obvious to me that that must be the case. Why, for instance, could P not be jerked towards Q when the string snaps taut so that P then catches up with Q some time later? One important consequence of P and Q travelling at the same speed after the string has gone taut (if that's what they do) is that kinetic energy is lost somewhere. KE can't be expended in the string, which is light and inextensible, so it must be expended in momentary deformation of P and Q. I won't bore everyone with the working here, but if P and Q were to interact without loss of KE, then after the initial jerk when the string goes taut, P would have to be travelling at 6u/5 and Q would have to be travelling at u/5.

The question would be fine if it explicitly stated that P and Q would remain the same distance apart after the string initially goes taut, but it doesn't state that.