@ computer .

Let's be serious !!

First of all , the analogy by which u are drawing conclusion is ENTIRELY FALSE . That's because in the merry - go-round u don't have any appreciable gravitational force towards it . But in the problem u HAVE TO CONSIDER the gravitational force on the helicopter or whatever .

So now explain ur reasoning .

I think  now it would be clear to u why I was telling to solve the differential eqn of motion . Got it ?

Now the problem can be formulated as follows:

Consider a ball given an upwards velocity v wrt earth which is rotating at a constant angular speed w. Find the time when the ball hits the Earth again .Also find the angular displacement of the ball on hitting the earth .

If earth moves by same amount by the same time then it will land at the same point , otherwise not .

Do u agree with this formulation ? If agree then plz go ahead using spherical polar co-ordinates . That's not  so difficult but may be time-consuming ,

Anyway , I am pleased with ur clear-cut comment . An academic comment , I believe( Unlike some experts ),should be made like this .