Snowball Hill: Accretion-Coupled Rolling Growth (2D)

Hill & Snow

Slope angle θ 25°

Snow depth h 8 cm

Snow/ball density ρ 400 kg/m³

Ball & Rolling

Initial radius r₀ 10 cm

Pickup stickiness k 0.60

Rolling friction μᵣ 0.030

Live Readout

Distance rolled

0.0 m

Speed v

0.00 m/s

Radius r

10.0 cm

Mass m

1.68 kg

√s growth check

0.0%

Elapsed time

0.0 s

How it works

Rolling down the slope, the ball sweeps a strip of ground snow of width 2r and depth h. A fraction k of that swept volume sticks, so the ball's volume grows as dV = k·2r·h·ds. Since V = (4/3)πr³, dV = 4πr²dr, which gives a closed-form ODE:

dr/ds = kh / (2πr)   ⇒   r(s) = √(r₀² + (kh/π)·s)

That's the real "snowball effect": radius grows with the square root of distance travelled, not linearly. The panel's growth-curve check compares the simulated r(s) against this closed-form solution every frame.

The accreted snow starts at rest, so picking it up is inelastic and costs kinetic energy — exactly like a raindrop absorbing still air, or a variable-mass rocket in reverse. Energy balance for a solid-sphere rolling without slipping (KE = 7/10·mv², I = 2/5·mr²) gives the coupled equation of motion:

dv/dt = (5/7)g(sinθ − μᵣcosθ) − (6/7)(ṁ/m)v
ṁ = dm/ds · v = 2ρkrh·v

The (6/7)(ṁ/m)v term is real accretion drag: the faster the ball gains mass relative to its own mass, the more it is slowed by absorbing stationary snow — independent of ordinary rolling friction.

Slope too shallow for this friction — μᵣcosθ ≥ sinθ, the ball won't gain speed.