WALL OF FIRE
One speed, every angle — the shots fill a region capped by a parabola, y = v²/2g − gx²/2v². Anything above it is safe, forever.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A cannon fixed at one muzzle speed, free to swing to any angle. Fire it a thousand ways and the shells fill the sky — but not ALL of the sky. There is a curtain overhead they can never reach, and it has a definite shape.
The fan
Every launch angle traces its own parabola — low and flat, high and steep. Two different angles even share a landing spot: a shot at θ and one at 90°−θ carry the same range, one lobbing high, one whipping low.
The frontier
Take the outer edge of that whole fan of parabolas and it is itself a parabola — the parabola of safety. It peaks at v²/2g, exactly the height of a shot fired straight up, and meets the ground at v²/g, the 45° maximum range.
Tangent
This high-angle physical flight reaches its envelope tangency before landing: same height and slope at one point.
Audit
Audited: the envelope is the maximum over complete projectile parabolas. High-angle physical arcs touch it in flight; low-angle arcs land first and touch only on their below-ground analytic continuation. Complementary angles still share range exactly.