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That's a question from previous Physics tests which I am using to study.

I would say that no, if the acceleration of the object is constant, its velocity, which is directly related to the acceleration, will also remain the same and make the object always go on the same direction. Is my way of thinking correct?

Please correct me if I am wrong and give me better explanations if possible.

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If you throw a ball in space it will travel in a straight line. It you throw a ball on earth where it feels gravity (a constant acceleration) then it will follow a curved line instead. – Pricklebush Tickletush Jun 15 '13 at 17:46
Got it already, it can indeed change its direction but only one time. Considering, for example, throwing a object in the air on the Earth, it will start with a velocity (with opposite direction to the Earth) and acceleration $-9.8m/s^2$ (gravity), its velocity will be slowly decreased till it reach its maximum distance from the earth and $V=0$, then it's velocity becomes "negative", it changes its direction, and it goes towards the Earth, with increasing velocity. – Luan Cristian Thums Jun 15 '13 at 18:01
@Luan No, that is not correct. An object with constant acceleration can change its direction of motion continuously. The parabolic motion of a simple projectile is precisely this, as Alec said. – Mark Eichenlaub Jun 15 '13 at 18:16
@MarkEichenlaub can you explain me more about how it changes its direction constantly? With a practical example if possible, sorry, I don't see it. – Luan Cristian Thums Jun 15 '13 at 18:26
I already described a practical example - throw a ball. It experiences constant acceleration (downward) while moving in a parabola. – Mark Eichenlaub Jun 15 '13 at 19:03

Acceleration is the rate of change of velocity with respect to time. A object undergoing a constant acceleration has a constantly changing velocity. If the acceleration always points in the same direction as the velocity, the trajectory will be no different than if there were no acceleration, but in general acceleration will change the trajectory.

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But wasn't the question about constant acceleration? – Paul J. Gans Jun 16 '13 at 1:51
@PaulJ.Gans: Yes? Does my answer not address that somehow? – Dan Jun 16 '13 at 7:12

Yes it can. The requirement of your question is "constant acceleration". Parabolic motion is the example of constant acceleration with changing trajectory. Both the acceleration and the direction are constant which is around 9.8 m/s$^2$ and perpendicular with earth surface.

Parabolic Motion

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The answer to your question is yes, there can be a trajectory change if there is a corresponding force acting on the body such that there is a conservation of energy. The change would also manifest itself as a zero placement in that the object would not experience a change in motion.

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