Galileo's Parabola and Where It Breaks
Galileo proved the parabola in Two New Sciences and, in the very same book, named its limit: air. Launch an ideal ball and a real one from the same barrel and watch the gap open — a shorter range, a steeper descent, the apex pushed past the halfway mark, and the error growing with speed exactly as he warned it would.
How to use this simulation
- Press Launch both to fire the vacuum ball and the real one together
- Switch the projectile between a shot put, a football and a ping-pong ball
- Load Galileo's warning — the same ball at 55 m/s — to see the error grow
- Adjust the drag slider to sweep from vacuum through to heavy air resistance
What to look for
- The real ball always lands short, and the gap widens as the launch speed rises
- Its descent is visibly steeper than its ascent, so the path is no longer symmetric
- The apex sits past the halfway point of the range instead of exactly on it
- A shot put barely departs from the parabola while a ping-pong ball abandons it entirely
- In the vacuum setting the two paths lie exactly on top of one another
The physics behind it
The parabola follows from assuming the only force is a constant weight. Air adds a drag force that opposes the velocity and grows with it, roughly as v² at everyday speeds, so it is not constant and not vertical. That breaks the symmetry the parabola depends on: the ball loses horizontal speed throughout, so the descending half is steeper and shorter than the ascent and the apex shifts beyond the midpoint. Because drag grows faster than the velocity itself, the discrepancy widens sharply at higher launch speeds and for light objects with a large area for their mass.