Cup Toss Probability Lab

A tosser flicks a ping-pong ball at water cups. Every release has tiny errors - see how they become a landing scatter and a hit probability.

Presets

Target

Release (tosser)

Add release noise to single throws (batch throws always do)

Release noise (1 sigma)

World

Display

Hit rate

Rim-plane crossing points (top view, cm from target)

in cupmisscapture circle

Range error at rim height (cm)

Hit rate vs. throws (95 % Wilson interval)

Trajectory of the last throw (side view)

simulateddrag-free parabola

Energy of the ball (mJ)

kineticpotentialtotal

Drag and table/cup impacts remove energy; Magnus lift and wind redirect it. In a vacuum with no impacts the total stays constant.

1 - Does the Monte-Carlo agree with exact statistics?

Vacuum, one ideal cup, noise sigma: speed 1.5 %, elevation 0.8 deg, azimuth 0.8 deg. For three arcs the aim is solved; 5000 simulated throws each are compared with a deterministic integration of the exact drag-free range map over the same Gaussian errors (probability that the ball centre crosses the rim plane within 25.5 mm of the cup axis).

2 - Which arc is best?

Current Setup (noise, air, surface) with a single cup at the target. The aim is re-solved for every elevation from 25 to 80 deg and 800 noisy throws are simulated per angle; dots are the full simulation, the line is the drag-free ideal-capture estimate for the same noise.

3 - Direct arc or one-bounce shot?

Same aim solver and noise for both styles (bounce randomness from the surface's restitution scatter), 1500 throws each, compared with a two-proportion z-test.

The science

Flight. The ball (2.7 g, r = 20 mm, hollow shell so I = 2/3 m r²) obeys m a = m g + F_D + F_L. Drag F_D = -1/2 ρ C_d A |v-w| (v-w) with C_d = 0.45; Magnus lift F_L = 1/2 ρ A C_L |v-w|² (ω×v)/|ω×v| with C_L = 1/(2 + v/(rω)). Integrated with RK4 (0.5-2 ms steps).

Vacuum reference. Without air, height y = y₀ + v sinθ t - g t²/2, so the descending crossing of the rim plane happens at t = [v sinθ + √(v²sin²θ + 2g(y₀-H))]/g and the horizontal distance is v cosθ t. For a level shot R = v² sin 2θ / g.

Error propagation. A relative speed error shifts the range by δR/R ≈ 2 δv/v, an elevation error by δR = 2 v² cos 2θ δθ / g and an azimuth error moves it sideways by R δφ. The landing scatter is therefore roughly an ellipse; a steep arc has a large vertical entry angle, so the target looks bigger but the range is more sensitive to speed.

Hit probability. P(hit) = ∫∫∫ 1[crossing within r_c of the axis] N(v) N(θ) N(φ). Experiment 1 evaluates this integral on a grid for the exact vacuum map and compares it with Monte-Carlo; the binomial 95 % interval is the Wilson score interval.

Impacts. Impulse model for a sphere: j_n = -(1+e) v_n m, tangential impulse limited by Coulomb |j_t| ≤ μ j_n (or the sticking value), applied to velocity and spin (angular impulse r×J, I = 2/3 m r²). The cup is a revolved thick line (wall + rolled lip) so rim bounces and wall-assisted drops come from geometry, not a dice roll. A ball whose centre sinks 30 mm below the rim inside the cup counts as in the water.

Simplifications. Rigid, immovable cups (heavy with water); no air torque on the spin; constant C_d; table edges are not modelled; the noise on elevation, azimuth, speed, spin and hand position is independent Gaussian.