sigh let's talk about the fly first
we assume a fly in vacuum somehow managing to hover(which alone
defies physics) in midair when the train starts moving. In this
case, the fly is no different from a fly that's outside the train.
It's relative velocity to the train will increase as the train
accelerates away. Long story short, the fly will eventually hit the
back of the train.
If there is air however, and we assume an airtight train, ie,
closed system, the same fly will never hit the back of the train.
Because the fly is in contact with the train via the medium of air,
it will move along with the train.
The question of the boy jumping up on a moving train is even
simpler, and has nothing to do with projectile motion. To an
observer standing beside the tracks, the boy and the train are
travelling at the exact same horizontal velocity before the boy
jumps.
When the boy jumps, all force exerted by him on himself (actually,
to the train first via newton's third law) is assumably purely on
the vertical axis. Since there is no net force on the horizontal
axis, his velocity remains constant. Since velocity of boy in
horizontal direction matches the train, he will move the exact same
distance in the direction of travel the train takes.
Obviously if the train changes it's horizontal acceleration while
the boy is in midair, the boy will land on a different spot.
As for the helicopter hovering, hovering for half a day in midair
will not put it around the world. If that were true, we could
travel to Europe by jumping on the spot for a couple of hours.
Someone mentioned on the first page this phenomena was not inertia.
In actuality, this is exactly what it is.
