Gist: Near the surface everything falls with the same acceleration, m/s², independent of mass and shape.
Why it matters: A special case of constant acceleration where the constant is handed to you by the planet. The independence from mass and shape is an experimental fact, not something derived.
Can it be proved?
The lecture’s answer: “Not through first principle.”
Detail: Lecture 2, p.5 and p.6
Related: Kinematic equations, Measuring g by free fall, Physics