Rope Swing Release Lab

Swing a robot across a canyon, let go at the right moment. Where is the best place to release - and how hard does the rope pull?

Presets

Release

Swing

Canyon

World

Display

Rope tension (kN)

tensionrope ratingweight

Energy of the rider (J)

kineticpotentialtotal

Last attempt

1 - Which release angle gives the longest jump?

Level landing ground at the platform height, same rope, start angle and gravity as in Setup. Part A has no air drag so it can be checked against an exact formula; part B repeats it with your air-drag settings.

2 - Period and tension vs start angle

Measured swing period and peak rope tension, against the exact large-angle period T = 4 sqrt(L/g) K(sin(th0/2)), the small-angle formula 2 pi sqrt(L/g), and T_max = m g (3 - 2 cos th0).

3 - Does a heavier rider jump further?

Same swing and release, mass varied, with your air drag. Without drag mass cancels out completely.

The science

Swing. A rigid rope of length L (pivot to the rider's centre of mass) gives θ'' = -(g/L) sinθ - (ρ C_dA L / 2m) |θ'| θ'. This is solved without the small-angle approximation (RK4, 1 ms steps).

Period. For amplitude θ₀ the exact period is T = 4 √(L/g) K(sin(θ₀/2)) where K is the complete elliptic integral of the first kind (computed by the arithmetic-geometric mean). The familiar 2π√(L/g) is only the limit θ₀ → 0; at 120 degrees the real swing is 37% slower.

Speed and tension. Energy gives v² = 2 g L (cosθ - cosθ₀). The rope supplies the centripetal force: T = m (v²/L + g cosθ), so at the bottom T = m g (3 - 2 cosθ₀) - 2 weights for a swing from 60 degrees, 3.5 weights from 120. A rope that cannot take it snaps and the rider leaves on the tangent.

Release. After release the rider is a projectile launched tangent to the circle, at height y_r above the landing level with speed v(θ_r). Range x = L sinθ_r + v cosθ_r t, with t = [v sinθ_r + √(v² sin²θ_r + 2 g (y_r - y_land))]/g. Releasing late gives speed but a steep path; releasing early gives a flat path but low speed. The best angle is a compromise, well below 90 degrees, and depends on how far below the release point the ground is - it is not simply 45 degrees.

Gravity cancels. Both v² and the flight time scale with g, so without drag the range is the same on the Moon as on Earth, only slower. Air drag breaks that symmetry.

Simplifications. Point mass at the CoM; rigid massless rope; no swing-out of the rider's body; drag is C_dA v² (CdA 0.9 m² default); landing ends the run; the rider is an abstract robot.