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For a projectile launched from ground, am I correct in saying momentum is conserved in the $x$ direction because no forces act in the $x$ direction (ignoring air resistance), but momentum is not conserved in the $y$ direction because there is gravity acting vertically, so the net external force on the system is non zero in the y axis and the system is nonisolated in the y axis? But the system is isolated in the x axis.

So when we use momentum conservation do we have to apply it separately to each axis (x,y,z)? Is this what is meant by the term 'linear' momentum?

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Yes, you are right. You can treat the $x$ and $y$ dimensions separately.

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