Cali’s Cannon – Projectile Motion Game

Aim Cali’s cannon and find out how a thrown ball really moves. Cali has 12 challenges in 4 levels for you: hit a target, fire from a tower, predict where the ball will land, get it over a wall, fire on the Moon and beat air resistance. Every flight leaves a path with a dot for each tenth of a second, and you can measure it: flight time, range, highest point, velocity and forces. Your stars are saved in this browser.

How to play

• Challenges: Cali gives 12 challenges in 4 levels: hit targets, predict where the ball lands, clear a wall, fire on the Moon and beat the air. A challenge opens when you have solved the one before it; solved ones can be played again in any order.
• Aim and fire: set the angle or the speed that the challenge leaves free: use the − and + buttons or the slider, or drag the barrel. Then press Fire! A setting with a small lock is fixed by the challenge.
• Predict first: in some challenges you work out the flight time, the range or the height and type it before you fire. A number within 2 % of the exact value counts, with a decimal point or a comma. Fire! opens when your answer is right, and the flight confirms it.
• Measure: every path has a dot for each 0.1 s and a star at its highest point. Tap a dot to read the time, distance, height and velocity at that moment. The tape measure shows the distance between any two points; its ends snap to the dots and to the ground.
• Vectors and time: switch on the velocity arrow with its two components, or the forces: gravity and air drag. Pause, slow motion (× 0.25) and Step let you follow a flight bit by bit.
• Stars: the fewer shots you need, the more stars you get. A hint costs one star.
• Keyboard: ↑ ↓ change the angle, ← → the speed (with Shift by 1 m/s), Space or Enter fires, A switches the air resistance in challenge 10, P pauses, E erases the paths and Esc returns to the map.

What the game teaches

Two motions at once. A launched ball moves sideways at a steady speed while gravity pulls it down, so the two motions can be worked out separately: x = v₀ cos θ · t and y = h + v₀ sin θ · t − gt² / 2. Here v₀ is the initial speed, θ the launch angle, h the height of the cannon and g the acceleration of gravity.

Dropped and fired. A ball fired sideways and a ball dropped from the same height land at the same moment: the horizontal speed does not change how fast a ball falls. The falling time from a height h is t = √(2h / g).

Angle and range. From the ground the range is R = v₀² sin 2θ / g and the highest point is H = v₀² sin² θ / (2g). So without air the longest range is at 45°, and two angles that add up to 90° land at the same spot.

Air resistance. Air resistance shortens every flight and moves the best angle below 45°. Large, light balls are slowed the most: a beach ball stops after a few metres, a cannonball hardly notices the air.

The game uses metres (m), seconds (s), metres per second (m/s) and kilograms (kg), as in physics class. The acceleration of gravity is 9.81 m/s² on the Earth and 1.62 m/s² on the Moon. The ball leaves from the small cross on the cannon, so the formulas hold exactly.