Projectiles: Two Motions That Ignore Each Other
A thrown ball is two easy problems wearing a trench coat: steady sideways motion, plus plain free fall. Split them and everything computes.
Builds on: 0.1 Position & Velocity: Saying Where, Precisely0.2 Acceleration: When Velocity Itself Changes
The great decoupling
Throw a ball and its path looks complicated — a curve through the air, position and speed changing everywhere. The trick that tames it is almost suspiciously simple: horizontal and vertical motion run independently.
- Horizontally nothing pushes or pulls (ignoring air), so the sideways velocity just… stays. Steady motion, last lesson’s easiest case.
- Vertically the ball is in ordinary free fall: gravity feeds it −9.81 m/s of vertical speed every second, whether or not it also moves sideways.
The curved path — a parabola — is what you get when a steady march and an accelerating fall are stapled together. The bullet-drop version: fire a bullet horizontally and drop one from the same height at the same instant, and they hit the ground together. The fired one just lands very far away.
Splitting a launch
A launch at speed v and angle θ is handled by splitting v into components — the sideways share and the upward share:
The vertical share buys airtime: the ball rises until gravity has eaten all of v_y, then falls symmetrically. The horizontal share spends that airtime covering ground. Range is the product of the two — which is why 45° is the ideal compromise on flat ground (in a vacuum): tilt higher and you gain airtime but waste speed going up; flatter and you fly fast but land too soon.
Where the ideal breaks
Real projectiles feel air resistance, which steals range and drags the ideal angle below 45° — footballs and javelins do best around 35–40°. The parabola is the clean limit, and knowing the clean limit is what lets you see what air is doing: the difference between the model and the throw is the drag.
Classic physics puzzle: a hunter aims a tranquiliser dart directly at a monkey in a tree; the monkey lets go the instant the dart leaves. Where should the hunter have aimed? Exactly where they did: dart and monkey both fall with the same g from the moment of release, so gravity bends the dart precisely onto the falling monkey. Independence of motions, dressed as a joke.
⚗️ Lab — The Launcher
Angle, speed, fire. The trajectory is drawn with its two shadows: one falling, one marching.
- At the same speed, compare 30° and 60° — same range, different flight (the two halves of the 45° compromise).
- Find the angle that maximises range, then the one that maximises hang time.
- Watch the vertical shadow: it is exactly last lesson’s drop, no matter the angle.
Work it by hand — 0 / 2
No multiple choice here: compute the value and type it. Suffixes like 2.5k or 20m are understood; answers within ±3% count.