Question

Why are vertical and horizontal motion considered independent?

Answer 1

We know that in motion problems, displacement, velocity and acceleration all are vector quantities. Only for the sake of simplicity these are written as `s,v and a`.
For sake of completeness these should be written as `vecs, vecv and` `veca.`

We also know that for vector quantities, orthogonal component is always equal to zero as shown below.

For any vector `vecX`, component in the direction which makes an angle `theta` with the vector is given by `vecX cos theta`.
It follows that for `theta=90^@`, (i.e., in the perpendicular direction)
`cos90^@=0`
`=>vecX cos 90^@=0`

As such vertical and horizontal components of motion can be treated independently.