The Sprinkler's Secret Pattern
A rotating garden sprinkler throws water along a parabola — so why does the water pile up in one ring instead of spreading out evenly?
Introduction
A single jet of water launched at speed v and angle θ lands a distance R(θ) = v²sin(2θ)/g away — ordinary projectile motion. But a real oscillating sprinkler sweeps θ back and forth, so every instant it's throwing water at a different range. That turns a simple parabola into a probability distribution: where does the water actually end up? The easy answer assumes a smooth, continuous sweep. The more interesting answer is that real "impact" sprinklers click through a ratchet, one discrete angle at a time — turning the question into a genuine discrete-vs-continuous distribution problem, with water piling up wherever the range function R(θ) goes flat. Deriving the exact landing-distribution formula analytically sits at the AA HL ceiling (continuous random variables, topic 4.14); at SL, or without the calculus, simulate many discrete sprinkler clicks numerically and read the distribution straight off the histogram.
Guiding Questions
- If the sprinkler sweeps θ at a constant angular speed, is the water distributed evenly across the lawn?
- What happens to the range R(θ) near θ = 45°, and why might water pile up there?
- How would you model the landing distance as a continuous random variable versus a discrete one for a ratchet-click sprinkler?
- Does adding wind resistance change where the water piles up?
- Could you test your model against a real sprinkler's watering pattern?