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In order to understand this problem, we must make some assumptions. If we first assume that the rate of fuel consumption is constant and if we further assume that the spacecraft is experiencing a positive thrust due to its fuel consumption we can analyze the motion.
In order to leave the surface of the earth, the constant thrust of the rocket engine must be greater than g at the surface of the earth: it must be greater than 9.81 m/s^2. The net acceleration of the spacecraft will be the differnce in the acceleration from the thrust and that of g.
According to Newton's Univseral Law of Gravity, as the spaceship gets further from the earth, the acceleration due to gravity (g) will decrease, thus the difference between the acceleration due to thrust and that of gravity gets larger and therefore the spacecraft will experience a greater net acceleration.
We should also notice that the weight of the spacecraft will reduce as the spacecraft burns fuel. The result will be that the same thrust will produce a greater acceleration. Again the spacecraft will receive an even greater acceleration.
Third, we know that the density of air decreases as the spacecraft gets higher into the atmosphere. Therefore the same thrust will more efficiently move the spacecraft.
The result of these three changes will cause the spacecraft will continue at ever greater accelerations until it leaves the atmosphere.
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