Projectile Motion Lab

Calculus 3 · Axiom Academy

One vector function runs every shot, every arc, every launch — position, velocity, acceleration. Put r(t), v(t), and a(t) in your hands. You launch a ball at 64 ft/s , gravity pulls it down at g = 32 ft/s² , and a hoop sits 90 ft downrange — three things one vector function r(t) lets you do: fly it , read its velocity , and aim it . Pick a launch angle, hit Launch, and watch the ball trace its path. The arrow from the origin is the position vector r(t) — its tip is where the ball is at time t. Notice the red acceleration arrow: it never changes. Differentiate the position: v(t) = r′(t) . Drag time t along the arc and watch the velocity vector — it always points along the path (tangent). Its horizontal part never changes; its vertical part shrinks, hits zero at the top, then reverses. On level ground the range is R = v₀² sin(2θ) / g . Sweep the launch angle and watch the range curve fill in — where does the ball land farthest? The peak isn't at the steepest angle. It's exactly at 45°. One vector function, differentiated twice: r(t) tells you where it is, v(t) = r′(t) how fast and which way, a(t) = v′(t) = ⟨0, −g⟩ — constant, straight down. That same r → v → a chain runs a satellite's orbit, a drone's flight path, and a roller coaster's track. Get the position vector, and calculus hands you the rest.

This is the written version of the interactive lesson above. See the full Calculus 3 course.