Tetherball

What is conserved, and what is not

Energy is conserved. Angular momentum about the pole is not: the rope's line of action misses the axis by exactly one pole radius, so the tension exerts a torque of N·m when kg·m²/s of angular momentum is left at two metres of rope, rising to N·m at half a metre. Across the wrap the ball keeps % of the angular momentum it started with, a drop of %, while its kinetic energy changes by a factor of . The measured rate of loss agrees with −m a v²/r to .

The control: the same rope hauled through a hole

Drill a hole down the axis of the pole and haul the rope through it at the very same rate. The two set-ups look almost identical and behave in opposite ways. Now the pull is central, so angular momentum is conserved — to over the same integration — and the hauling does work on the ball. With m of rope left, the wrapping ball is still doing m/s and the hauled ball is doing m/s: times faster, times the kinetic energy, and J put in by whoever is pulling.

This is the contrast Kleppner and Kolenkow set as Problem 6.13, and it is exactly where the folk claim comes from: a textbook that treats the pole as a point gets the hole answer, because for a point pole the two problems are the same problem.

Why “just shrink the radius” is not a rope

The obvious shortcut is to keep the pivot on the pole's axis and let the radius shrink by one pole circumference per turn. That model keeps the speed constant too — its spread is — so speed alone cannot tell the two apart, and it finishes the wrap only parts per million later ( s against s), so timing barely separates them either. What separates them completely is the direction of the force. The shortcut needs a pull that points % off the rope line at full extension and ° off it near the end. No rope can pull sideways. In the real wrap the constraint force lies along the rope to .

In three dimensions the ball rises, so it slows down

A real rope hangs from the top of the pole and wraps as a helix. Working that out exactly gives a small surprise: the ball's height is H − L·sin γ, where γ is the angle the free rope makes below the horizontal and L is the whole rope, not the part still in the air. Height therefore does not depend on how much rope is wrapped, and energy conservation reads v² = v₀² + 2gL(sin γ − sin γ₀) exactly — checked against the integrator to m/s. The ball speeds up if and only if it descends. Started from a steady cone and wrapped to half a metre of rope, it instead rises m and loses % of its speed, from to m/s, with energy held to .

Nothing in the physics makes the ball faster. Only the players do.

A published answer we could not reproduce

David Morin's Introduction to Classical Mechanics sets the tetherball as Problem 5.54 and prints the ratio of final to initial speed as sin θ₀, where θ₀ is the angle the rope makes with the pole at launch. Two pages earlier the same chapter works the reverse process — a mass released at the pole, letting the rope unwind — and gets tan θ = √2, an angle of °. We reproduce that unwinding angle exactly from his own two equations.

Running those same two equations forwards through the winding process gives v_f/v_i = √(1 − 2cot²θ₀), which is zero at ° — as time reversal of his own example demands, since a ball launched at the critical angle must arrive at the pole at rest. The printed answer is there instead of zero, and too large at 60°. Taken at face value at 60° it would create J out of nothing for this ball, against mgL = J. We report the disagreement rather than resolve it: the two answers agree only at 90°, a flat horizontal circle.

How the numbers on this page were produced

An offline harness runs assertions in sections against the same engine the game runs, and writes every figure above into a generated file. None of them is typed by hand.

• The wrapped path is the involute of the pole's circle. Against Wikipedia's parametric form it agrees to m, and the documented arc length (r/2)t² reproduces our arc coordinate to m.
• Two integrators that share no code — an arc-length reduction and a general constraint-manifold RK4 — agree on the ball's position to m.
• RK4 converges at order over steps of s (errors ).
• Touchdown events are located by bisection, not stepped past. Located, the terminal error converges at order ; stepped past, at order — averaged over targets so the grid cannot alias it. At the finest step that is s against s, a factor of .
• The three-dimensional map's analytic derivatives match finite differences to and over states; |∂B/∂γ| = L to ; the mass matrix is diagonal to ; the height identity holds .
• deliberately broken variants were run to confirm the checks can fail, and of them turned out inert. Moving the rope's departure point one part in ten thousand off the tangency circle raises the no-work cosine to ; switching air drag on drops the speed to of its start, so the constant speed above is a result and not a tautology.
• Air drag alone sets the speed: m/s² at m/s, a half-speed time of s, matching the closed form to . Friction between the rope and the pole cannot slow the ball at all: whatever its size, it is transmitted along the rope, and the rope is perpendicular to the ball's motion.
• The match runs a second, closed-form integrator of its own — the exact solution rather than a discretisation — so that is pinned to the analytic one directly: over combinations of speed, step and drag it reproduces the closed-form arc to and the speed to , one long step equals nine hundred short ones to , and the event locator inverts it to m over cases.
• seeded matches are pinned outcome by outcome, which is how we found that a replay is exact within one JavaScript engine and not across two: shifting Math.atan2 by a single unit in the last place — the difference we measured between two versions of the same engine — reaches the ball after physics steps ( s) and turns a s match into a s one, % longer.
• self-play matches at equal skill: the first player wins % ± %, median length s, ninetieth percentile s, ended by a completed wrap and on the clock. The busiest match used physics steps against a hard cap of . Raising one side's skill wins % of .

Every figure, as measured

About this tetherball

Tetherball is a folk playground game with no single inventor, no publisher and no owner. A tethered-ball variant was described by Jessie H. Bancroft in 1909, and schools and parks took up the modern game in the 1920s. The name is descriptive, not a trademark; the patents that exist cover apparatus, not the game — US 2,496,795 (Johnson, 1950), US 3,397,887 (Caplan, 1968) and US 4,248,423 (Lotfy, 1981).

This is an independent reimplementation. No code, art, sound or data comes from any existing tetherball product. The simulation, the renderer and the opponent were all written from the documented rules and from the mechanics of a rope wrapping a cylinder.

What is faithful

• A ft pole with a in outer diameter, a ft tether, and a volleyball-sized ball ( kg, m radius) hanging about m above the ground at rest.
• A ft circular court split by a line through the pole, with each player confined to their own half.
• The game is won by wrapping the rope completely in your own direction — turns on these dimensions, which is m of travel for the ball.
• Swinging at a ball that is in your opponent's half is a foul and costs you a longer wait.

What differs, and why

• The view is from above and the simulation is planar. The game plays the horizontal-plane tetherball problem, which is the version that has an exact solution and the one the textbooks set. The three-dimensional helical wrap is solved exactly too, but it lives in the physics section rather than in the match.
• The coils are drawn wider than they are. A m pole radius against a m court would make the wrap invisible, so the drawn rope thickens outward as it winds. Every number in the simulation uses the true radius.
• Hands are points with a reach, not arms. You set an angle and swing; contact quality depends on how centred the ball is in your reach and how early in the swing it arrives.
• No height, no serve rule, no double-hit rule. The 5 ft mark that a real winning wrap must clear ( m) has no meaning in a plan view, and the four-revolution serve rule is not universal enough to be worth imposing.
• A stalled ball is re-served rather than left hanging, and a match that reaches the clock is awarded to whoever has more wrap.
• A seeded match replays exactly on your machine, not on everyone's. JavaScript does not specify the last bit of its trigonometric functions, and the opponent's aim goes through one of them every physics step; see the physics section for how far one bit travels.
• Air drag is modelled; rope friction is not — because, as the physics section shows, rope friction cannot change the ball's speed.

Provenance of the numbers

Pole height, tether length, resting height, the winning-wrap rule and the 5 ft mark are documented. The pole diameter, the court diameter and the ball's mass and size are qualified: they come from retailer specifications and from the volleyball the ball is described as resembling, because tetherball has no governing body and no rule book gives them. The opponent's reaction, reach and swing timing are reconstructed to make a playable match and are not claimed to be measurements of anything.

Controls

Mouse or arrow keys to place your hand; click or Space to swing; N for a new game; P to pause. On a touchscreen, tap the court to place your hand and swing in one move.