Curling Physics

What is modelled

A full curling sheet (4.75 m wide, hog line to hog line 21.95 m, 12-foot house) and 19.1 kg granite stones. You choose the release speed, aim and spin; the stone then glides on pebbled ice, curls, can be swept, and can hit the stones already in play. Eight stones make an end; the model scores it the way the rules do (stones in the house closer to the button than the opponent's best).

Equations

• Friction on pebbled ice falls with speed: μ(v) = μ1 / √v, deceleration a = μ g. Then v3/2 decreases linearly in time, the stone stops after t = v03/2 / (1.5 μ1 g) and travels x = 0.4 v05/2 / (μ1 g). With μ₁ = 0.0125 a hog-line release of 2.4 m/s reaches the far tee.
• Sweeping warms the ice film and lowers friction: μ → μ (1 − 0.2 s), so full sweeping lets the same stone travel 25 % farther.
• Curl: a clockwise stone drifts right, a counter-clockwise one left. The model uses a sideways acceleration alat = c μ g tanh(ω/0.3) (1 − 0.35 s) perpendicular to the motion.
• Spin decays by dω/dt = −0.1 μ g rb / k2 with running-band radius r_b = 0.065 m and radius of gyration k = 0.1 m.
• Stone-stone hits: equal masses, smooth contact, normal restitution e = 0.85: J = (1+e) m vn / 2 and kinetic energy loss (1 − e2) m vn2 / 4. The tangential motion is untouched.

Checks run in the page

• Stopping distance and time of a spinless stone against the closed forms.
• Work-energy theorem (friction work equals the initial kinetic energy).
• Sweeping extends the run by exactly 1/0.8.
• Collision: momentum, tangential speed and the kinetic-energy loss formula.
• Curl mirror symmetry and the number of turns.

Honest simplifications

• Why a curling stone curls is still debated in the research literature (pivot-slide, scratch-guide and thin-film models); the sideways force here is an empirical law tuned to about 1 m of curl on a draw, not a derivation from first principles.
• The 1/√v friction form, the 20 % sweeping effect, the spin-decay coefficient and e = 0.85 are fitted, representative values; real ice varies with pebble, temperature and the stone's own running band.
• Stones are 2D discs with no tangential contact friction, so spin is not transferred in a hit.
• The release is from the near hog line; the delivery slide is not modelled.

• Throw a draw with the default settings, then press Aim at button after changing spin or ice: the aim is solved by bisection on the same physics.
• Hold the sweep button while the stone is between the hog lines: sweeping cuts friction by up to 20 percent and keeps the stone straighter and longer.
• Throw two stones at the house and a fast takeout: the collision loses energy as (1 - e squared) times the normal relative speed squared over four.