Run Detail

completedclaude-opus-5.5

95excellent

Every model's answer from this round

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Provider

openrouter

Challenge Version

v1.0

Turns

Duration

Batched — not timed

Total Tokens

60,130

Input Tokens

673

Output Tokens

59,457

Reasoning Tokens

44,350

Reasoning Effort

default

Est. Cost

$0.5959

Started

10/26/2026, 1:55:38 PM

OR Provider

Anthropic

Finish Reason

stop

Generation ID

gen-batch-1793022937-0f6c31272fd3b709f9be

Turns (1)

Turn #1

673 in59,457 out44,350 reasoning60,130 total

user

You build physics toys, and you have dropped a garden hose with the water on full.

Build me a single HTML page: a garden hose lying on a lawn, the spigot on, and the nozzle released.

The build

A hose of fixed length, pinned at the spigot at one end, free at the nozzle end. It bends, carries mass, and does not stretch.
Water pressure control shown on screen in psi, adjustable from 0 to full and back down.
Water leaving the nozzle pushes back on it. At low pressure the hose settles. Past some pressure it starts to whip across the lawn. Lowering the pressure settles it again.
Water particles exit along the current nozzle vector and land on the ground.
Clicking and dragging pins that point of the hose to the mouse; releasing frees it.
Loading the page as #psi=40 sets the pressure to 40 psi with the nozzle free on load. Any number works.
The art is yours.

House rules

One file. Everything inline — no CDN, no external stylesheet, font, script or image. data: URIs are fine.
No libraries. No physics engine — write your own.
No network at runtime.
No localStorage, sessionStorage, cookies or indexedDB — sandboxed frame, they throw.
Return the file as exactly one fenced html block.

Answer in exactly these sections

The model

One paragraph, before the file: how the hose discretization and inextensibility constraints are implemented, the formula for the nozzle reaction force as a function of psi, and ground friction handling.

The prediction

The pressure in psi at which the hose transitions from stationary or settling to continuous whipping, as one number, derived from your model parameters.

The page

The single fenced html block.

What it actually does

Where the motion looks like a real hose and where it does not. Name any artificial damping, injected noise or scripted oscillation. Everyone has watched a hose get away from somebody; say what yours gets wrong before I see it.

assistant

The model

The hose is 3.0 m long with mass 0.45 kg/m (rubber plus the water inside) and bending stiffness EI = 2.0 N·m². It is split into N = 30 rigid links of l = 0.1 m. The generalized coordinates are the absolute link angles θᵢ, so the hose cannot stretch by construction; there is no constraint solver to drift or leak energy. Masses are lumped at the link ends, with half mass at the tip. Lagrange's equations give a dense mass matrix M_ij = l²·S_max(i,j)·cos(θᵢ−θⱼ), where S_j is the mass beyond link j. Each RK4 stage solves M·θ̈ = Q − Σⱼ l²S_max(i,j) sin(θᵢ−θⱼ) θ̇ⱼ² by Cholesky, with a timestep of at most 1/1500 s. Bending is a rotational spring EI/l at every joint. The spigot is a clamp: position and outgoing direction are fixed, with a spring EI/l against the fixed spout angle. "Pinned at the spigot" therefore means clamped, which is how a hose threaded onto a bib behaves.

The nozzle is a 4 mm orifice. With Bernoulli exit speed v = √(2p/ρ) and an ideal jet, the reaction is

F = ρAv² = 2·p·A = 2·(psi·6894.76 Pa)·π(0.002 m)² ≈ 0.1733 N per psi.

That force is applied at the tip node opposite the current tip tangent. It is a follower load, so the model is Beck's column.

Ground friction is not Coulomb. It is an isotropic linear viscous drag of −γ·m·v on every node, with γ = 1.2 s⁻¹. Because it is mass-proportional, it reduces exactly to −γθ̇ in the angle coordinates. There is no stiction and no static friction cone.

The prediction

27.5 psi

Derivation:

Undamped threshold. A clamped-free rod under a tangential tip load flutters at F·L²/EI = 20.05 (Beck).
Damping shift. The dimensionless damping is c = γ·L²·√(m/EI) = 1.2 · 9 · 0.474 = 5.12.
- A two-mode Galerkin model uses cantilever modes with λ⁴ = 12.36 and 485.5, and bᵢⱼ = [[0.858, 1.874], [−11.74, −13.29]].
- With mass-proportional damping, an undamped complex eigenvalue ω² = a ± ib becomes a growing mode only when b² > a·c².
- That condition pushes the threshold up by 1.39, giving F·L²/EI ≈ 21.44.
Convert to force. F_cr = 21.44 · 2.0 / 9 = 4.76 N.
Convert to pressure. p = F/(2A) = 1.896·10⁵ Pa = 27.5 psi.

The expected onset frequency is about 11/t₀ = 2.6 rad/s, a period of roughly 2.4 s.

The page computes this same number live from its own parameters and shows it in the panel. Discretization at N = 30 should shift it by a percent or two; I have not measured that.

The page

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1">
<title>Dropped garden hose</title>
<style>
html,body{margin:0;height:100%;overflow:hidden;background:#3d6a24;font-family:system-ui,-apple-system,"Segoe UI",Roboto,sans-serif;user-select:none;-webkit-user-select:none}
#c{position:fixed;left:0;top:0;width:100%;height:100%;touch-action:none;cursor:grab}
#c.grabbing{cursor:grabbing}
#panel{position:fixed;left:12px;top:12px;width:262px;background:rgba(18,26,16,.84);color:#e9f1e2;border-radius:10px;padding:11px 13px 10px;font-size:12px;box-shadow:0 4px 18px rgba(0,0,0,.35);line-height:1.35}
#panel h1{font-size:13px;margin:0 0 4px;font-weight:650;letter-spacing:.02em;color:#cfe8b8}
.psirow{display:flex;align-items:baseline;gap:6px}
#psiBig{font-size:34px;font-weight:700;font-variant-numeric:tabular-nums;color:#fff}
.u{font-size:14px;color:#a9c79a}
#psi{width:100%;margin:4px 0 6px;accent-color:#e07b2a}
.btns{display:flex;gap:4px;align-items:center}
button{background:#33462b;color:#eaf5df;border:1px solid #577a45;border-radius:5px;padding:3px 7px;font-size:12px;cursor:pointer}
button:hover{background:#46603a}
#psiNum{width:62px;background:#1d2a19;color:#fff;border:1px solid #577a45;border-radius:5px;padding:2px 4px;font-size:12px}
table{width:100%;border-collapse:collapse;margin:7px 0 5px}
td{padding:1px 0}
td:last-child{text-align:right;font-variant-numeric:tabular-nums;color:#fff}
#status{font-weight:700}
#chart{width:236px;height:64px;display:block;margin:3px 0 5px;background:#10180d;border-radius:4px}
label{display:block;margin:3px 0;color:#c9dcbc}
.hint{color:#9fb893;margin:5px 0 0;font-size:11px}
#slow{color:#ffb35c;display:none}
</style>
</head>
<body>
<canvas id="c"></canvas>
<div id="panel">
  <h1>Garden hose, dropped, water on</h1>
  <div class="psirow"><span id="psiBig">0</span><span class="u">psi</span><span id="slow">slow-mo</span></div>
  <input id="psi" type="range" min="0" max="100" step="0.5" value="0">
  <div class="btns">
    <button data-d="off">Off</button><button data-d="-5">−5</button><button data-d="5">+5</button><button data-d="full">Full</button>
    <input id="psiNum" type="number" min="0" step="0.5" value="0">
  </div>
  <table>
    <tr><td>jet reaction F</td><td id="rF">0</td></tr>
    <tr><td>jet speed / flow</td><td id="rV">0</td></tr>
    <tr><td>load F·L²/EI</td><td id="rP">0</td></tr>
    <tr><td>predicted onset</td><td id="rPred">–</td></tr>
    <tr><td>hose kinetic energy</td><td id="rKE">0</td></tr>
    <tr><td>status</td><td id="status">–</td></tr>
  </table>
  <canvas id="chart"></canvas>
  <label><input type="checkbox" id="noise" checked> flow-turbulence noise (2% side force)</label>
  <button id="relay">Re-lay hose</button>
  <p class="hint">Drag any point of the hose to hold it there; release to let go. ↑/↓ = ±1 psi. Chart: orange = psi (dashed = predicted onset), cyan = log KE.</p>
</div>
<script>
'use strict';
(function(){
const PSI_PA=6894.757;
const P={L:3.0,N:30,m:0.45,EI:2.0,dNoz:0.004,rho:1000,gamma:1.2,thetaBase:0.35,
         kPin:800,cPin:30,full:100,noiseFrac:0.02,noiseTau:0.05};
const N=P.N,l=P.L/N,l2=l*l,kb=P.EI/l,A=Math.PI*(P.dNoz/2)*(P.dNoz/2);

// lumped node masses (node k = end of link k-1), tip half mass
const mk=new Float64Array(N+1);
for(let k=1;k<=N;k++) mk[k]=P.m*l*(k===N?0.5:1);
const S=new Float64Array(N);
for(let j=0;j<N;j++){let s=0;for(let k=j+1;k<=N;k++)s+=mk[k];S[j]=s;}

const th=new Float64Array(N),om=new Float64Array(N);
const Cc=new Float64Array(N),Ss=new Float64Array(N),M=new Float64Array(N*N),R=new Float64Array(N),Q=new Float64Array(N);
const a1=new Float64Array(N),a2=new Float64Array(N),a3=new Float64Array(N),a4=new Float64Array(N);
const t2=new Float64Array(N),t3=new Float64Array(N),t4=new Float64Array(N);
const w2=new Float64Array(N),w3=new Float64Array(N),w4=new Float64Array(N);

let psi=0,Fjet=0,Fn=0,grab=-1,gx=0,gy=0,noiseOn=true;

function accel(t,w,out){
  for(let i=0;i<N;i++){Cc[i]=Math.cos(t[i]);Ss[i]=Math.sin(t[i]);}
  const cN=Cc[N-1],sN=Ss[N-1];
  let d=t[0]-P.thetaBase;
  for(let i=0;i<N;i++){
    const dn=i<N-1?t[i+1]-t[i]:0;
    let q=-kb*d+kb*dn;                            // bending
    q-=l*Fjet*(sN*Cc[i]-cN*Ss[i]);                // follower jet reaction, -F*tangent at tip
    q+=l*Fn*(cN*Cc[i]+sN*Ss[i]);                  // lateral turbulence force at tip
    Q[i]=q;d=dn;
  }
  if(grab>0){
    let x=0,y=0,vx=0,vy=0;
    for(let i=0;i<grab;i++){x+=l*Cc[i];y+=l*Ss[i];vx-=l*w[i]*Ss[i];vy+=l*w[i]*Cc[i];}
    const fx=P.kPin*(gx-x)-P.cPin*vx,fy=P.kPin*(gy-y)-P.cPin*vy;
    for(let i=0;i<grab;i++)Q[i]+=l*(-Ss[i]*fx+Cc[i]*fy);
  }
  for(let i=0;i<N;i++){
    let h=0;const ci=Cc[i],si=Ss[i],row=i*N;
    for(let j=0;j<N;j++){
      const wij=l2*S[i>j?i:j];
      M[row+j]=wij*(ci*Cc[j]+si*Ss[j]);
      h+=wij*(si*Cc[j]-ci*Ss[j])*w[j]*w[j];
    }
    R[i]=Q[i]-h;
  }
  // Cholesky (lower, in place)
  for(let j=0;j<N;j++){
    let s=M[j*N+j];
    for(let k=0;k<j;k++)s-=M[j*N+k]*M[j*N+k];
    const dd=Math.sqrt(s);M[j*N+j]=dd;
    for(let i=j+1;i<N;i++){
      let v=M[i*N+j];
      for(let k=0;k<j;k++)v-=M[i*N+k]*M[j*N+k];
      M[i*N+j]=v/dd;
    }
  }
  for(let i=0;i<N;i++){let s=R[i];for(let k=0;k<i;k++)s-=M[i*N+k]*out[k];out[i]=s/M[i*N+i];}
  for(let i=N-1;i>=0;i--){let s=out[i];for(let k=i+1;k<N;k++)s-=M[k*N+i]*out[k];out[i]=s/M[i*N+i];}
  for(let i=0;i<N;i++)out[i]-=P.gamma*w[i];      // mass-proportional ground drag
}
function step(h){
  accel(th,om,a1);
  for(let i=0;i<N;i++){t2[i]=th[i]+0.5*h*om[i];w2[i]=om[i]+0.5*h*a1[i];}
  accel(t2,w2,a2);
  for(let i=0;i<N;i++){t3[i]=th[i]+0.5*h*w2[i];w3[i]=om[i]+0.5*h*a2[i];}
  accel(t3,w3,a3);
  for(let i=0;i<N;i++){t4[i]=th[i]+h*w3[i];w4[i]=om[i]+h*a3[i];}
  accel(t4,w4,a4);
  for(let i=0;i<N;i++){
    th[i]+=h/6*(om[i]+2*w2[i]+2*w3[i]+w4[i]);
    om[i]+=h/6*(a1[i]+2*a2[i]+2*a3[i]+a4[i]);
  }
}
function relay(){
  for(let i=0;i<N;i++){th[i]=P.thetaBase+0.9*Math.sin(1.25*Math.PI*i/(N-1));om[i]=0;}
  Fn=0;
}
function gauss(){const u=1-Math.random(),v=Math.random();return Math.sqrt(-2*Math.log(u))*Math.cos(2*Math.PI*v);}

// ---- prediction from the same parameters (2-mode Galerkin + damping, shifted to exact Beck 20.05)
function predict(){
  const L4=[Math.pow(1.8751041,4),Math.pow(4.6940911,4)];
  const B=[[0.8582,1.8738],[-11.7432,-13.2943]];
  const t0=P.L*P.L*Math.sqrt(P.m/P.EI),c=P.gamma*t0;
  function unstable(p,cc){
    const k11=L4[0]+p*B[0][0],k12=p*B[0][1],k21=p*B[1][0],k22=L4[1]+p*B[1][1];
    const tr=k11+k22,det=k11*k22-k12*k21,disc=tr*tr-4*det;
    if(disc>=0)return (tr-Math.sqrt(disc))/2<0;
    return (-disc/4)>(tr/2)*cc*cc;
  }
  function crit(cc){let p=0;while(p<300&&!unstable(p,cc))p+=0.05;let lo=Math.max(0,p-0.05),hi=p;
    for(let i=0;i<50;i++){const mm=(lo+hi)/2;if(unstable(mm,cc))hi=mm;else lo=mm;}return hi;}
  const pnd=crit(c)-(crit(0)-20.05);
  return {pnd,psi:pnd*P.EI/(P.L*P.L)/(2*A)/PSI_PA,c,t0};
}
const pred=predict();

// ---- particles
const MAXP=6000;
const pX=new Float32Array(MAXP),pY=new Float32Array(MAXP),pZ=new Float32Array(MAXP),
      pVX=new Float32Array(MAXP),pVY=new Float32Array(MAXP),pVZ=new Float32Array(MAXP),pK=new Float32Array(MAXP);
let np=0,emitAcc=0;
function tipState(){
  let x=0,y=0,vx=0,vy=0;
  for(let i=0;i<N;i++){const c=Math.cos(th[i]),s=Math.sin(th[i]);x+=l*c;y+=l*s;vx-=l*om[i]*s;vy+=l*om[i]*c;}
  return [x,y,vx,vy,Math.cos(th[N-1]),Math.sin(th[N-1])];
}
function emit(tb,ts,vj){
  if(np>=MAXP)return;
  const el=0.12+Math.random()*0.08,az=(Math.random()-0.5)*0.05;
  const ca=Math.cos(az),sa=Math.sin(az);
  const dx=ts[4]*ca-ts[5]*sa,dy=ts[5]*ca+ts[4]*sa;
  const v=vj*(0.86+0.14*Math.random()),vh=v*Math.cos(el);
  const vx=dx*vh+ts[2],vy=dy*vh+ts[3],vz=v*Math.sin(el);
  pX[np]=ts[0]-vx*tb;pY[np]=ts[1]-vy*tb;pZ[np]=0.035-vz*tb;
  pVX[np]=vx;pVY[np]=vy;pVZ[np]=vz;pK[np]=0.15+Math.random()*0.55;np++;
}

// ---- canvas / view
const cv=document.getElementById('c'),ctx=cv.getContext('2d');
const lawn=document.createElement('canvas'),wet=document.createElement('canvas'),wctx=wet.getContext('2d');
let W=0,H=0,dpr=1,scale=100,ox=0,oy=0;
const X=x=>ox+x*scale,Y=y=>oy+y*scale;
function resize(){
  dpr=Math.min(2,window.devicePixelRatio||1);W=innerWidth;H=innerHeight;
  cv.width=Math.round(W*dpr);cv.height=Math.round(H*dpr);
  scale=Math.min(W/10.5,H/7.4);ox=W>760?W*0.57:W*0.5;oy=H*0.47;
  makeLawn();
  wet.width=cv.width;wet.height=cv.height;wctx.setTransform(dpr,0,0,dpr,0,0);
}
function makeLawn(){
  lawn.width=cv.width;lawn.height=cv.height;
  const g=lawn.getContext('2d');g.setTransform(dpr,0,0,dpr,0,0);
  const grd=g.createRadialGradient(ox,oy,40,ox,oy,Math.max(W,H)*0.85);
  grd.addColorStop(0,'#5d9135');grd.addColorStop(1,'#3a6522');
  g.fillStyle=grd;g.fillRect(0,0,W,H);
  g.save();g.translate(W/2,H/2);g.rotate(-0.25);
  const band=Math.max(40,scale*0.6),Rr=Math.hypot(W,H);
  let i=0;for(let x=-Rr;x<Rr;x+=band,i++){g.fillStyle=i%2?'rgba(255,255,220,0.05)':'rgba(0,25,0,0.05)';g.fillRect(x,-Rr,band,2*Rr);}
  g.restore();
  let seed=12345;const rnd=()=>{seed=(Math.imul(seed,1664525)+1013904223)>>>0;return seed/4294967296;};
  const cols=['#2f5a1b','#3b6b22','#4a7d2b','#5a8f35','#6aa03e','#7bb048','#34601f','#86b850'];
  const count=Math.floor(W*H/55);
  g.lineWidth=1;g.globalAlpha=0.55;
  for(let c=0;c<cols.length;c++){
    g.strokeStyle=cols[c];g.beginPath();
    for(let n=0;n<count/cols.length;n++){
      const x=rnd()*W,y=rnd()*H,len=2+rnd()*5,a=-Math.PI/2+(rnd()-0.5)*1.3;
      g.moveTo(x,y);g.lineTo(x+Math.cos(a)*len,y+Math.sin(a)*len);
    }
    g.stroke();
  }
  g.globalAlpha=1;
  for(let n=0;n<W*H/9000;n++){
    const x=rnd()*W,y=rnd()*H;
    if(rnd()<0.3){g.fillStyle='rgba(250,215,50,0.9)';g.beginPath();g.arc(x,y,1.7,0,7);g.fill();}
    else{g.fillStyle='rgba(125,175,85,0.55)';for(let k=0;k<3;k++){g.beginPath();g.arc(x+Math.cos(k*2.1)*2,y+Math.sin(k*2.1)*2,1.8,0,7);g.fill();}}
  }
}

// ---- UI
const slider=document.getElementById('psi'),num=document.getElementById('psiNum'),big=document.getElementById('psiBig');
const rF=document.getElementById('rF'),rV=document.getElementById('rV'),rP=document.getElementById('rP'),
      rPred=document.getElementById('rPred'),rKE=document.getElementById('rKE'),rSt=document.getElementById('status'),
      slowEl=document.getElementById('slow');
rPred.textContent=pred.psi.toFixed(1)+' psi';
function setPsi(v){
  v=Number(v);if(!isFinite(v))v=0;psi=Math.max(0,v);
  if(psi>Number(slider.max))slider.max=String(Math.ceil(psi));
  slider.value=String(psi);num.value=String(Math.round(psi*10)/10);
  big.textContent=psi<100?psi.toFixed(1):psi.toFixed(0);
}
slider.addEventListener('input',()=>setPsi(slider.value));
num.addEventListener('change',()=>setPsi(num.value));
document.querySelectorAll('button[data-d]').forEach(b=>b.addEventListener('click',()=>{
  const d=b.getAttribute('data-d');
  if(d==='off')setPsi(0);else if(d==='full')setPsi(P.full);else setPsi(psi+Number(d));
}));
document.getElementById('noise').addEventListener('change',e=>{noiseOn=e.target.checked;});
document.getElementById('relay').addEventListener('click',relay);
addEventListener('keydown',e=>{
  if(e.target&&e.target.tagName==='INPUT'&&e.target.type==='number')return;
  if(e.key==='ArrowUp'){setPsi(psi+1);e.preventDefault();}
  else if(e.key==='ArrowDown'){setPsi(psi-1);e.preventDefault();}
});
function readHash(){
  const m=/psi\s*=\s*([-+]?(?:\d+\.?\d*|\.\d+)(?:e[-+]?\d+)?)/i.exec(location.hash||'');
  if(m)setPsi(parseFloat(m[1]));
}
addEventListener('hashchange',readHash);

// pointer: pin nearest node to mouse
let mouseX=0,mouseY=0;
function worldOf(e){const r=cv.getBoundingClientRect();return [(e.clientX-r.left-ox)/scale,(e.clientY-r.top-oy)/scale];}
cv.addEventListener('pointerdown',e=>{
  const [wx,wy]=worldOf(e);let x=0,y=0,best=-1,bd=Infinity;
  for(let k=1;k<=N;k++){x+=l*Math.cos(th[k-1]);y+=l*Math.sin(th[k-1]);const d=Math.hypot(x-wx,y-wy);if(d<bd){bd=d;best=k;}}
  if(bd<Math.max(0.3,26/scale)){grab=best;gx=wx;gy=wy;mouseX=wx;mouseY=wy;cv.setPointerCapture(e.pointerId);cv.classList.add('grabbing');}
});
cv.addEventListener('pointermove',e=>{if(grab>0){const [wx,wy]=worldOf(e);gx=wx;gy=wy;mouseX=wx;mouseY=wy;}});
function release(){grab=-1;cv.classList.remove('grabbing');}
cv.addEventListener('pointerup',release);cv.addEventListener('pointercancel',release);

// chart
const chart=document.getElementById('chart'),cctx=chart.getContext('2d');
chart.width=236*dpr;chart.height=64*dpr;cctx.setTransform(dpr,0,0,dpr,0,0);
const HN=400,hKE=new Float32Array(HN).fill(-6),hPsi=new Float32Array(HN);let hHead=0,hAcc=0;
function drawChart(){
  const w=236,h=64,g=cctx;g.clearRect(0,0,w,h);
  const pmax=Math.max(100,Number(slider.max));
  g.strokeStyle='rgba(255,160,60,0.55)';g.setLineDash([3,3]);g.beginPath();
  const yp=h-2-(pred.psi/pmax)*(h-4);g.moveTo(0,yp);g.lineTo(w,yp);g.stroke();g.setLineDash([]);
  g.lineWidth=1.3;
  g.strokeStyle='#ff9a3c';g.beginPath();
  for(let i=0;i<HN;i++){const v=hPsi[(hHead+i)%HN],x=i/(HN-1)*w,y=h-2-(v/pmax)*(h-4);i?g.lineTo(x,y):g.moveTo(x,y);}g.stroke();
  g.strokeStyle='#5fe0ff';g.beginPath();
  for(let i=0;i<HN;i++){const v=hKE[(hHead+i)%HN],x=i/(HN-1)*w,y=h-2-((v+6)/8)*(h-4);i?g.lineTo(x,y):g.moveTo(x,y);}g.stroke();
}

// ---- drawing
const sx=new Float64Array(N+1),sy=new Float64Array(N+1),wxN=new Float64Array(N+1),wyN=new Float64Array(N+1);
function hosePath(dx,dy){
  ctx.beginPath();ctx.moveTo(sx[0]+dx,sy[0]+dy);
  for(let k=1;k<N;k++)ctx.quadraticCurveTo(sx[k]+dx,sy[k]+dy,(sx[k]+sx[k+1])/2+dx,(sy[k]+sy[k+1])/2+dy);
  ctx.lineTo(sx[N]+dx,sy[N]+dy);
}
let wetFrame=0,splats=[];
function render(){
  ctx.setTransform(dpr,0,0,dpr,0,0);
  ctx.drawImage(lawn,0,0,W,H);
  ctx.drawImage(wet,0,0,W,H);
  wxN[0]=0;wyN[0]=0;
  for(let k=0;k<N;k++){wxN[k+1]=wxN[k]+l*Math.cos(th[k]);wyN[k+1]=wyN[k]+l*Math.sin(th[k]);}
  for(let k=0;k<=N;k++){sx[k]=X(wxN[k]);sy[k]=Y(wyN[k]);}
  const hw=Math.max(6,0.045*scale);
  const cb=Math.cos(P.thetaBase),sb=Math.sin(P.thetaBase);
  const rx=X(-cb*0.13),ry=Y(-sb*0.13);

  // riser shadow + spout
  ctx.fillStyle='rgba(0,0,0,0.3)';ctx.beginPath();ctx.ellipse(rx+5,ry+6,0.07*scale+3,0.06*scale+3,0,0,7);ctx.fill();
  ctx.lineCap='round';
  ctx.strokeStyle='#7d8288';ctx.lineWidth=hw*0.85;ctx.beginPath();ctx.moveTo(rx,ry);ctx.lineTo(sx[0],sy[0]);ctx.stroke();

  // hose
  ctx.lineJoin='round';
  ctx.strokeStyle='rgba(0,0,0,0.28)';ctx.lineWidth=hw+3;hosePath(3,4);ctx.stroke();
  ctx.strokeStyle='#1f5e22';ctx.lineWidth=hw+1.5;hosePath(0,0);ctx.stroke();
  ctx.strokeStyle='#2f8a33';ctx.lineWidth=hw;hosePath(0,0);ctx.stroke();
  ctx.strokeStyle='rgba(230,215,70,0.85)';ctx.lineWidth=Math.max(1,hw*0.13);hosePath(hw*0.12,hw*0.12);ctx.stroke();
  ctx.strokeStyle='rgba(255,255,255,0.28)';ctx.lineWidth=Math.max(1,hw*0.28);hosePath(-hw*0.2,-hw*0.22);ctx.stroke();

  // nozzle (last 16 cm drawn over the hose)
  const tc=Math.cos(th[N-1]),ts=Math.sin(th[N-1]);
  const nl=0.16*scale,bxn=sx[N]-tc*nl,byn=sy[N]-ts*nl;
  ctx.lineCap='butt';
  ctx.strokeStyle='rgba(0,0,0,0.3)';ctx.lineWidth=hw*1.35;ctx.beginPath();ctx.moveTo(bxn+3,byn+4);ctx.lineTo(sx[N]+3,sy[N]+4);ctx.stroke();
  ctx.strokeStyle='#9c7420';ctx.lineWidth=hw*1.35;ctx.beginPath();ctx.moveTo(bxn,byn);ctx.lineTo(sx[N],sy[N]);ctx.stroke();
  ctx.strokeStyle='#e2b94c';ctx.lineWidth=hw*0.55;ctx.beginPath();ctx.moveTo(bxn-ts*hw*0.2,byn+tc*hw*0.2);ctx.lineTo(sx[N]-ts*hw*0.2,sy[N]+tc*hw*0.2);ctx.stroke();
  ctx.strokeStyle='#2a2a2a';ctx.lineWidth=hw*1.45;ctx.beginPath();ctx.moveTo(bxn,byn);ctx.lineTo(bxn+tc*hw*0.35,byn+ts*hw*0.35);ctx.stroke();
  ctx.strokeStyle='#c99b33';ctx.lineWidth=hw*0.8;ctx.beginPath();ctx.moveTo(sx[N],sy[N]);ctx.lineTo(sx[N]+tc*hw*0.5,sy[N]+ts*hw*0.5);ctx.stroke();

  // coupling + riser + handle
  ctx.fillStyle='#c9a13c';ctx.beginPath();ctx.arc(sx[0],sy[0],hw*0.75,0,7);ctx.fill();
  ctx.fillStyle='#8f9499';ctx.beginPath();ctx.arc(rx,ry,0.055*scale,0,7);ctx.fill();
  ctx.fillStyle='#b9bec3';ctx.beginPath();ctx.arc(rx-1.5,ry-1.5,0.035*scale,0,7);ctx.fill();
  const ha=psi/P.full*4.5,hr=0.085*scale;
  ctx.strokeStyle='#b8231d';ctx.lineWidth=Math.max(2.5,0.018*scale);
  ctx.beginPath();ctx.arc(rx,ry,hr,0,7);ctx.stroke();
  ctx.beginPath();for(let k=0;k<4;k++){const a=ha+k*Math.PI/2;ctx.moveTo(rx,ry);ctx.lineTo(rx+Math.cos(a)*hr,ry+Math.sin(a)*hr);}ctx.stroke();
  ctx.fillStyle='#7a1410';ctx.beginPath();ctx.arc(rx,ry,0.02*scale,0,7);ctx.fill();

  // water drops: shadows then elevated streaks
  if(np){
    ctx.fillStyle='rgba(0,20,10,0.13)';ctx.beginPath();
    for(let i=0;i<np;i++){ctx.rect(X(pX[i])-1,Y(pY[i])-1,2,2);}
    ctx.fill();
    ctx.strokeStyle='rgba(215,238,255,0.78)';ctx.lineWidth=Math.max(1.2,scale*0.011);ctx.lineCap='round';ctx.beginPath();
    const zf=0.8*scale;
    for(let i=0;i<np;i++){
      const x=X(pX[i]),y=Y(pY[i])-pZ[i]*zf;
      ctx.moveTo(x,y);ctx.lineTo(x-pVX[i]*0.018*scale,y-pVY[i]*0.018*scale+pVZ[i]*0.018*zf);
    }
    ctx.stroke();
  }

  // grab indicator
  if(grab>0){
    ctx.strokeStyle='rgba(255,255,255,0.8)';ctx.lineWidth=1.5;ctx.setLineDash([4,3]);
    ctx.beginPath();ctx.moveTo(sx[grab],sy[grab]);ctx.lineTo(X(mouseX),Y(mouseY));ctx.stroke();ctx.setLineDash([]);
    ctx.fillStyle='rgba(255,230,200,0.85)';ctx.beginPath();ctx.arc(X(mouseX),Y(mouseY),7,0,7);ctx.fill();
    ctx.strokeStyle='rgba(80,40,20,0.8)';ctx.stroke();
  }
  // scale bar
  ctx.fillStyle='rgba(255,255,255,0.85)';ctx.fillRect(W-20-scale,H-24,scale,3);
  ctx.font='11px system-ui,sans-serif';ctx.fillText('1 m',W-20-scale,H-30);
}

// ---- main loop
let last=performance.now(),keSm=0,hud=0;
function frame(now){
  let dt=(now-last)/1000;last=now;if(!(dt>0))dt=0;dt=Math.min(dt,1/30);
  Fjet=2*psi*PSI_PA*A;
  const vj=Math.sqrt(2*psi*PSI_PA/P.rho),flow=A*vj;
  const wEst=Math.sqrt((16*kb+4*l*Fjet)/(P.m*l*l2));
  const hMax=Math.min(1/1500,1.2/wEst);
  let n=Math.max(1,Math.ceil(dt/hMax)),simDt=dt,slow=false;
  if(n>300){n=300;simDt=n*hMax;slow=true;}
  const h=simDt/n,rate=Math.min(3000,flow*3.0e6);
  for(let s=0;s<n;s++){
    if(noiseOn&&Fjet>0){Fn+=-Fn*h/P.noiseTau+P.noiseFrac*Fjet*Math.sqrt(2*h/P.noiseTau)*gauss();}else Fn=0;
    step(h);
    emitAcc+=rate*h;
    if(emitAcc>=1){const tsx=tipState(),tb=(s+1)*h;while(emitAcc>=1){emitAcc-=1;emit(tb,tsx,vj);}}
  }
  if(!isFinite(th[N-1])||!isFinite(om[N-1]))relay();
  slowEl.style.display=slow?'inline':'none';

  // particles
  const dtp=simDt;splats.length=0;
  for(let i=0;i<np;){
    let vx=pVX[i],vy=pVY[i],vz=pVZ[i];
    const sp=Math.sqrt(vx*vx+vy*vy+vz*vz),f=1/(1+pK[i]*sp*dtp);
    vx*=f;vy*=f;vz=vz*f-9.81*dtp;
    const x=pX[i]+vx*dtp,y=pY[i]+vy*dtp,z=pZ[i]+vz*dtp;
    if(z<=0||Math.abs(x)>40||Math.abs(y)>40){
      if(z<=0)splats.push(x,y);
      np--;pX[i]=pX[np];pY[i]=pY[np];pZ[i]=pZ[np];pVX[i]=pVX[np];pVY[i]=pVY[np];pVZ[i]=pVZ[np];pK[i]=pK[np];continue;
    }
    pX[i]=x;pY[i]=y;pZ[i]=z;pVX[i]=vx;pVY[i]=vy;pVZ[i]=vz;i++;
  }
  if(splats.length){
    wctx.fillStyle='rgba(18,45,28,0.16)';wctx.beginPath();
    const r0=Math.max(1,scale*0.012);
    for(let i=0;i<splats.length;i+=2){const x=X(splats[i]),y=Y(splats[i+1]),r=r0*(0.7+Math.random()*1.1);wctx.moveTo(x+r,y);wctx.arc(x,y,r,0,7);}
    wctx.fill();
  }
  if(++wetFrame%20===0){wctx.globalCompositeOperation='destination-out';wctx.fillStyle='rgba(0,0,0,0.04)';wctx.fillRect(0,0,W,H);wctx.globalCompositeOperation='source-over';}

  // kinetic energy
  let ke=0,vx=0,vy=0;
  for(let k=1;k<=N;k++){const i=k-1;vx-=l*om[i]*Math.sin(th[i]);vy+=l*om[i]*Math.cos(th[i]);ke+=0.5*mk[k]*(vx*vx+vy*vy);}
  keSm+=(ke-keSm)*Math.min(1,simDt/0.7);
  hAcc+=simDt;while(hAcc>=0.1){hAcc-=0.1;hKE[hHead]=Math.log10(ke+1e-6);hPsi[hHead]=psi;hHead=(hHead+1)%HN;}

  render();
  if(++hud%4===0){
    rF.textContent=Fjet.toFixed(2)+' N';
    rV.textContent=vj.toFixed(1)+' m/s · '+(flow*60000).toFixed(1)+' L/min';
    rP.textContent=(Fjet*P.L*P.L/P.EI).toFixed(1)+' (crit '+pred.pnd.toFixed(1)+')';
    rKE.textContent=ke<1e-3?(ke*1000).toFixed(2)+' mJ':ke.toFixed(3)+' J';
    let st,col;
    if(grab>0){st='held by you';col='#ffd27a';}
    else if(keSm<5e-4){st='settled';col='#9fe39a';}
    else if(keSm>0.05){st='whipping';col='#ff7a5c';}
    else{st='moving / settling';col='#ffe08a';}
    rSt.textContent=st;rSt.style.color=col;
    drawChart();
  }
  requestAnimationFrame(frame);
}

resize();addEventListener('resize',resize);
relay();readHash();
requestAnimationFrame(t=>{last=t;requestAnimationFrame(frame);});
})();
</script>
</body>
</html>

What it actually does

What I expect from the math (I have not timed it against the 27.5 psi prediction):

Below about 27 psi the hose rings down and settles. Near the threshold the decay is very slow, because the damping margin is small.
Above about 28 psi a sideways swing grows into a large, sweeping limit cycle.
- The growth is slow just above threshold and about one e-fold per second at 40 psi.
- The onset period is about 2.4 s.
- At high pressure the shape picks up higher-mode kinks near the nozzle, and the motion gets faster and more violent.
Lowering the pressure lets the motion die out again. I don't know whether the flutter onset is supercritical. If it isn't, there may be some hysteresis, with whipping persisting a little below the onset when you come down from high pressure. The chart lets you check.
Inextensibility is exact. The hose cannot stretch at all.
The jet always leaves along the current nozzle tangent plus the nozzle's own velocity. Watching the spray arcs smear as the nozzle swings is the most convincing part.

What is artificial:

Damping: an isotropic viscous drag of 1.2 s⁻¹ stands in for the lawn. There is no other damping. RK4 at dt ≤ 1/1500 s adds negligible numerical damping to the modes you can see.
Injected noise: a random lateral force at the nozzle with standard deviation 2% of the jet force and a 50 ms correlation time. It is on by default and has a checkbox.
- It exists because a perfectly straight hose above threshold would otherwise sit there until floating-point roundoff grew, which could take a minute or more.
- It also makes the hose quiver slightly below threshold.
No scripted oscillation. All motion comes from the follower load, the bending, the drag and the noise.

What it gets wrong, compared with the hose that got away from you:

No stiction or memory. A real hose on grass stays wherever it was dropped because of Coulomb friction and its coiled set. Mine has neither. At 0 psi it springs and slides back to a straight line out of the spigot within a few seconds. It glides on the lawn as if the lawn were syrup.
It stays flat on the ground. The real violent part is the nozzle lifting off and thrashing in 3D: rearing up, slapping down, spinning. Mine is a 2D rod and never leaves the ground plane. Only the water droplets have height.
It is too orderly and too slow. A real loose hose flails chaotically at several Hz. Mine settles into a fairly regular, snake-like swing with a period of seconds. To get even that, I made the hose shorter (3 m) and stiffer (EI = 2 N·m²) than a real slack 15 m hose.
Wrong fluid physics for a hose. Only the jet reaction is modeled, as a follower force (Beck's column). The water flowing inside the hose is left out:
- There is no Coriolis coupling and no flow-curvature term.
- Proper pipe-conveying-fluid theory, with water mass about 40% of the total, predicts a markedly higher and differently shaped instability boundary.
- There is no pressure drop along the hose, so spigot psi equals nozzle psi, which overstates the thrust.
No collisions. The hose passes through itself and through the spigot.
The grab is a spring, not a hard pin. Your hand is a stiff spring at 800 N/m, so the held point lags the cursor by a few centimeters under strong jet thrust.
The spigot is a clamp. It holds the outgoing direction as well as position, rather than being a free pin.
Arbitrary pressures run, but slowly. Very large #psi= values work, but the simulation falls into slow motion (flagged on screen) to stay stable.

Attachments

anthropic/claude-opus-5.5 — page