There's something deeply satisfying about watching a cannonball arc through the air and land exactly where you intended. That satisfaction becomes even richer when you realize you're not just playing a game—you're doing mathematics.
Parabolic Cannon is a browser-based game that transforms the abstract world of quadratic equations into something you can see, tweak, and immediately understand. No graph paper required.
The Core Idea
Every projectile follows a parabolic path. This is one of those beautiful intersections of physics and mathematics that we often teach separately but rarely let students experience together. In Parabolic Cannon, your goal is simple: adjust a quadratic equation so that the cannonball lands in a basket. Miss, and you'll see exactly how your equation translated into motion. Hit, and you'll have internalized something about parabolas that no textbook explanation could match.
The game gives you five tries per target. After each shot, you can see your previous trajectories ghosted on the screen—a visual history of your mathematical reasoning. Too high? Too far? The evidence is right there, guiding your next adjustment.
Three Ways to Think About the Same Curve
What makes this more than just a targeting game is the ability to switch between different representations of the same parabola:
Standard Form: y = ax² + bx + c
This is the form most students encounter first. The coefficient a controls how "steep" the parabola is (and whether it opens up or down—for our cannon, it must be negative). The coefficient b affects the initial direction, and c shifts everything vertically.
Playing with standard form, you develop intuition for questions like: "If I want the ball to travel farther, should I increase or decrease |a|?" The answer becomes obvious after a few shots in a way that solving equations on paper never quite achieves.
Vertex Form: y = a(x - h)² + k
Here's where things get interesting for students who've struggled with the abstraction of "completing the square." In vertex form, h and k directly specify the highest point of the trajectory. Want the cannonball to peak at x = 200 and reach a maximum height of 150? Just set h = 200 and k = 150.
This representation makes it trivial to aim for a specific apex—something that requires real algebraic manipulation to figure out from standard form. Playing with both forms builds an intuitive bridge between them.
Polar Form: r(φ) with Physics Parameters
The polar representation takes a different approach entirely. Instead of abstract coefficients, you specify:
- v₀: Initial velocity (how hard you're firing)
- θ: Launch angle in degrees
- g: Gravitational acceleration
This is the physicist's view of projectile motion. The game displays the computed range and maximum height, connecting the physical parameters to their geometric consequences. Students can explore questions like: "What launch angle maximizes range?" (Spoiler: it's 45° in ideal conditions, but seeing it emerge from experimentation is far more memorable than being told.)
Why Games Work for Math Education
There's substantial research behind game-based learning, but the intuition is simple: games provide immediate, meaningful feedback in a context where failure is cheap and iteration is natural.
When a student gets a wrong answer on a worksheet, the feedback is often delayed and abstract—a red mark, a point deduction. When a cannonball misses a basket, the feedback is instant and visceral. You see the error. More importantly, you see how it was wrong, which points directly toward correction.
The game also sidesteps the anxiety that many students feel around mathematics. You're not solving equations; you're aiming a cannon. The mathematics happens almost incidentally, which is often exactly what's needed to build genuine understanding before formal symbolic manipulation.
Technical Notes
Parabolic Cannon runs entirely in the browser—no installation, no plugins, just HTML, CSS, and JavaScript. The cannon's barrel angle adjusts in real-time as you modify your equation, providing continuous visual feedback. A dashed preview line shows your current trajectory before you fire, letting you reason about your parameters without committing to a shot.
The game tracks your score across rounds, with points awarded based on how few tries you needed. This adds a light competitive element without making it feel like a test.
Try It Yourself
Whether you're a student trying to build intuition for quadratic functions, a teacher looking for a classroom demonstration, or just someone who enjoys the intersection of math and games, Parabolic Cannon offers a few minutes of genuinely educational entertainment.
Load it up, adjust your coefficients, and fire away. The parabola will teach you the rest.
Parabolic Cannon was built as an experiment in making abstract mathematical concepts tangible. The best way to understand a parabola might just be to throw something along one.