Sure, let’s decompose the swing into a pendulum‑like system with some added horizontal motion. 1. **Basic pendulum motion** - Rope length \(L\). - Angle from vertical \(\theta\). - Tangential speed \(v_t = L\dot{\theta}\). - Radial (centripetal) acceleration \(a_r = \frac{v_t^2}{L} = L\dot{\theta}^2\). 2. **Energy conservation** \[ \frac{1}{2}m v_t^2 + mgL(1-\cos\theta)=\text{constant} \] If you release at an initial height \(h\) above the lowest point, then \[ v_t = \sqrt{2g h} \] with \(h = L(1-\cos\theta_{\text{max}})\). 3. **Horizontal launch toward the next building** At the lowest point you have a horizontal component \(v_x = v_t\). If the next building is at distance \(D\) and you want to land there, treat it as a projectile with initial speed \(v_x\) and gravity \(g\). Time of flight \(t = \frac{2v_x}{g}\) (assuming you launch and land at the same vertical level). Required horizontal distance \(D = v_x t = \frac{2v_x^2}{g}\). So solve for the needed initial speed: \[ v_x = \sqrt{\frac{D g}{2}} \] Then adjust the swing angle so that the tangential speed at the bottom equals this \(v_x\). 4. **Tension check** The maximum tension at the lowest point is \[ T_{\text{max}} = mg + \frac{m v_t^2}{L} = mg + \frac{m v_x^2}{L} \] Make sure your web material can handle this. 5. **Practical tweak** If the buildings differ in height, add a vertical component: \[ v_y = \sqrt{2g\Delta h} \] Then the launch speed becomes \(\sqrt{v_x^2 + v_y^2}\), and adjust the angle accordingly. In short: pick a rope length \(L\), compute the required bottom speed \(v_t\) to clear the horizontal gap \(D\) using \(v_t = \sqrt{D g / 2}\), ensure the tension \(T_{\text{max}}\) stays within material limits, and then launch. Adjust the initial release angle to hit that speed. That’s the core of the math; real‑world nuances (wind, rope elasticity, body position) will add extra terms, but this gives a solid baseline.

Nice summary, but don’t underestimate the wind’s impulse—add a drag term and maybe a small lift if the web flares. And always keep a safety factor on tension, just in case the rope flexes. Good luck with the calculations, and enjoy the flight!

Thanks, that’s the mindset—keep the calculations tight, the safety margin high, and the wind factor in the models. Stay curious and keep testing the numbers; that’s the real fun. Good luck!