TwiddleLab / Math / Algebra in Motion
Project: Hit the Target
What you will learn
- Where a shot lands is a root: solve c + bx − 0.05x² = height of the target.
- The highest point is the vertex, halfway along the arc.
- If b² − 4ac is negative, no tilt can reach the target. The math knows before you fire.
How the lesson goes
A cannon at the edge of a cliff. The target is on a floating island.
Tap the three Fire buttons. Each tilts the barrel a little more.
Level 1: hit the island by trial and error.
Turn the barrel's slope b and Fire. Your shot count is saved.
Predict before you fire: write the landing equation.
The ball lands on the island when its height equals the island's: 20 + bx − 0.05x² = 10. Move everything to one side.
Level 2: the island moved. Aim with the formula, fire once.
Drag the barrel's tip. The pin shows where the formula says the ball lands. The arc stays hidden.
Level 3: this island floats at 60 m. Can anything reach it?
The steepest barrel has b = 2. Drag the marker to the highest point you think the ball reaches.
Three questions, three tools.
Where does it land? Solve height = target. The roots from the quadratic formula are the landing points.
How high does it go? The vertex, halfway between the roots. Completing the square shows it.
Can it reach at all? Check b² − 4ac first. Negative means no tilt works.
Trial and error, or the formula?
Your shot counts from level 1 and level 2 side by side. Tap the method that works better for a new island: fewer shots, and it warns you when a target is out of reach.
Challenge: one arc through two hoops.
Set the slope b and the tower height c. Par is 3 shots.
Games move a ball frame by frame. Run this game loop.
Each frame the height changes by vy, and gravity takes 2 off vy. Press Run.
A ball flies along y = −x² + 4x + 5. At what positive x does it land?
Land means y = 0.
Make your card: My cannon.
Drag your level 2 arc and its formula into the frame.
Previous lesson: The Quadratic Formula · Part of the course: Algebra in Motion
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