# Basis of 2D motion analysis

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?

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:

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:

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

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.

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.