Understanding the Swing
This simulator models Spider-Man's web-swing as a two-part motion. First, the swing itself is a circular pendulum. The web attaches to the top of the building and Spider-Man starts from rest at the building's edge. He swings down and releases the web at a chosen angle relative to the horizontal. After release, he becomes a projectile, following a parabolic path until he lands on the ground (or a lower surface).
The math uses energy conservation to find the speed at release: potential energy lost equals kinetic energy gained. The formula is v = sqrt(2 * g * h_drop), where g is gravity (9.81 m/s²) and h_drop is the vertical distance the center of mass falls during the swing. That depends on web length and the angle at release.
Once released, we use standard projectile equations. The horizontal range is v * cos(θ) * t, where t is the total time in the air. The time is found from the vertical motion, accounting for the release height above ground.
Assumptions and Limitations
- The web is massless, rigid, and does not stretch.
- Air resistance is ignored.
- Spider-Man is treated as a point mass at the end of the web.
- The building is vertical and the ground is flat.
- The swing starts from rest at the building's top edge.
These simplifications make the problem solvable with high school physics. In reality, a human body has size, the web stretches, and wind matters. But this model gives a good first estimate and shows how changing inputs affects the outcome.
Try These Experiments
- Set the release angle to 0° (horizontal). You'll get the maximum range for a given height, but the hang time is short because the vertical velocity component is zero.
- Use a very long web with a high building. The speed at release can be huge, sending Spider-Man far across the city.
- Compare the Empire State Building preset with a custom lower height. Notice how the range changes.
Common Mistakes
One common error is thinking the release angle is measured from the vertical. It's from the horizontal, like a standard projectile problem. Another is forgetting that the release point is not at ground level—it's at the building height minus the vertical drop during the swing. That height affects how long Spider-Man stays airborne after letting go.
If the web length is greater than the building height, the simulation still works: the swing starts from a point above the building top, which isn't physically possible but lets you explore extreme cases.