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Only Gravitation force exist. Rocket dropped from height $h$ on ground. Without firing rocket can hit ground in $t$ second.Rocket has enough fuel for journey of $T$ time. Rocket does not have any kind of wings, so can't create upper lift force in presence of any fluid.

So will this rocket hit ground in same time ($t$) irrespective of that it was not fired or fired during journey?

Consider g=GM/r^2 (for any planet like earth)

enter image description here

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it'll hit the ground at the same time, the vertical acceleration is independent from the horizontal movement. (because it's a free fall it doesn't depend on the mass too, if you think that firing can change mass )

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  • $\begingroup$ Does this assume a flat earth (i.e. the distance traveled horizontally is much smaller than the curvature)? $\endgroup$
    – user1583209
    Commented Jan 13, 2017 at 11:23
  • $\begingroup$ yes, looking at your picture the ground is flat. $\endgroup$
    – Ismasou
    Commented Jan 13, 2017 at 11:24
  • $\begingroup$ (It's a simplification you set in your assumptions, if you want to ask about orbital or sub-orbital trajectories you can make another question.) $\endgroup$
    – JMLCarter
    Commented Jan 13, 2017 at 11:42
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Assuming a flat ground, no atmosphere, and no "tumbling" of the rocket under power, they would both hit the ground at the same time.

With an atmosphere and a rocket that has a cylindrical shape as you have drawn, the rockets will tend to rotate to keep their noses pointed toward their direction of travel. As a consequence, the thrust vector of the powered rocket will not stay parallel to the ground and the powered rocket will hit the ground first.

Things could get more complicated if the center of mass of the powered rocket is far enough toward the rear. Then the rocket would have a tendency to fall rear-first due atmospheric drag.

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