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The basic argument for analysing 2D motion is that if we have a projectile we can break its 2D motion into 2 1D motions along 2 perpendicular axes. The motions along these axes and their corresponding accelerations are independent of each other.

I was wondering whether this fact has a 'rigorous' proof or is based on experimentation? Could somebody please clarify?

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This is simply a conclusion of a vector algebra. Speed, acceleration and force are vectors, and vector can be decomposed into two projections along chosen any two directions in a plane containing this vector: enter image description here

So basically it's just a vector addition rule: $\vec{AB} = \vec{A} + \vec{B}$.

Now, when it comes to independent projections it means that decomposed vector projections are perpendicular to each other, like so:

enter image description here

$\vec{c} = \vec{a}_{\perp} + \vec{b}_{\perp}$

So vector decomposition into independent parts is a special case of general vector projections addition along arbitrary axis. This special perpendicular decomposition has deep roots in Pythagorean theorem.

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Using Newton's equation $\vec{F}=m\vec{a}$ for 2D motion, we remember that it is actually a vector equation. Hence, we can decompose it into 3 separate scalar equations.

$F_x=ma_x$ $F_y=ma_y$ $F_z=ma_z$

Where in the 2D case we can ignore the z-component.

Now comes the crucial step, where we want to argue that the x and y equation are independent and that we can treat them separately. We can only do this if $F_x$ is independent of $y$, and $F_y$ is independent of $x$. This is often the case, especially if you are only considering gravity, and the axes such that the force of gravity points in one of the directions.

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