A water jet from a nozzle under pressure: Bernoulli speed, flow and reaction force, the ballistic path with air drag on droplets, and the stagnation pressure on a soil bank.
The science
Nozzle. Bernoulli's equation between the pressurised hose and the open jet gives v = Cv√(2p/ρ), with velocity coefficient Cv = 0.97 (friction) and water density ρ = 1000 kg/m³. The flow is Q = Cd A v, A = πd²/4, Cd = 0.95 (contraction). The jet carries power ½ρQ v² (less than the supplied pQ because of nozzle losses) and pushes back on the monitor with F = ρQ v, which is why a monitor must be anchored or balanced.
Path. After a short coherent length the jet breaks into droplets; the model follows a representative drop of effective diameter dd with quadratic air drag: dv/dt = −g − (ρaCDAd/2m)|vrel|vrel, where CD = 0.45 for a sphere, vrel is relative to the air (including a head wind) and m is the droplet mass. Integrated with RK4 at 2 ms. Larger drops (a coherent jet behaves like a very large drop) keep their speed better; a thin spray falls short.
Range and angle. Without drag and from height h the range is R = (v/g)√(v²+2gh) at the best angle tanθ = v/√(v²+2gh); with drag the best angle drops below this. The graph shows both.
Bank. A jet that is stopped by a wall exerts a stagnation pressure ps = ½ρv² (v = the speed at the bank). If it exceeds the erosion threshold of the soil the bank is cut; a coherent jet loses speed with distance, here through the droplet model only, which is an optimistic simplification.
Numerical checks. (1) With drag off, h = 0 the RK4 range equals v²sin2θ/g. (2) With drag off and h > 0 the best angle found by scanning equals atan(v/√(v²+2gh)) and the range equals the formula. (3) Energy: with drag on, the kinetic-plus-potential energy lost between the nozzle and the ground equals the work done by drag along the path.
Honest simplifications. A real jet is a continuous core that breaks up gradually and entrains air, so a single droplet size is a rough model; the water hammer, pump curve, hose losses, soil mechanics (cohesion, grain size) and the cutting rate are not modelled. Pressures of 0.2–4 MPa and soil thresholds are illustrative values. Historical hydraulic mining caused severe environmental damage; the page is a physics demonstration only.