Projectile Path

Trigonometry · Axiom Academy

LESSON Projectile Path: Parametric Equations in Motion Explore how trigonometry describes the path of a projectile through parametric equations—separating horizontal and vertical motion. When you throw a ball, launch a rocket, or kick a soccer ball, the object follows a curved path called a parabola . But how do we mathematically describe where the object is at any given moment? Instead of using a single equation y = f(x), we use parametric equations that describe the horizontal position (x) and vertical position (y) separately as functions of time (t): v₀ = initial velocity (speed at launch) α = launch angle (measured from horizontal) g = gravitational acceleration (≈ 9.8 m/s²) The genius of parametric equations is that they separate horizontal and vertical motion , which follow different rules: → Horizontal Motion: x = v₀cos(α)t Moves at constant velocity because there's no horizontal force (ignoring air resistance). The cosine component gives the horizontal part of the initial velocity. ↑ Vertical Motion: y = v₀sin(α)t - ½gt² Starts with upward velocity (sine component) but gravity pulls it down , creating the curved path. The -½gt² term represents gravitational acceleration. By treating x and y independently, we can accurately predict where the projectile will be at any moment in time! Interactive Projectile Launcher

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