Before the object is launched vertically upward inside the train compartment
it has the same speed as the train.
So by the inertia law of Newton
it will continue forever with that velocity,
unless an external force acts on the object.
Now, being launched vertically upward, the object certainly experiences
an external force.
However, that force acts in the vertical direction
which is perpendicular to the horizontal motion of the particle.
As a consequence it cannot perform work in the horizontal direction.
The horizontal speed of the object is not influenced by the external force
which launches the object vertically upward.
Moreover, after being launched vertically upward the only external force
acting on the object is gravity, the weight of the object.
That force is also perpendicular to the horizontal motion of the object.
We may thus conclude that during its motion in the air
the object has all the time a constant horizontal speed
with respect to the observer on the platform of the station.
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