14.3: Spider-Man

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In this example we’ll develop a model of Spider-Man swinging from a springy cable of webbing attached to the top of the Empire State Building. Initially, Spider-Man is at the top of a nearby building, as shown in Figure 14.1.

14.1.jpg — Figure 14.1: Diagram of the initial state for the Spider-Man example

The origin, O, is at the base of the Empire State Building. The vector H represents the position where the webbing is attached to the building, relative to O. The vector P is the position of Spider-Man relative to O. And L is the vector from the attachment point to Spider-Man.

By following the arrows from O, along H, and along L, we can see that

H + L = P

So we can compute L like this:

L = P - H

Exercise 14.3

As an exercise, simulate this system and estimate the parameters that maximize the distance Spider-Man swings.

Implement a model of this scenario to predict Spider-Man’s trajectory.
Choose the right time for Spider-Man to let go of the webbing in order to maximize the distance he travels before landing.
Choose the best angle for Spider-Man to jump off the building, and the best time to let go of the webbing, to maximize range.

Use the following parameters:

According to the Spider-Man Wiki (https://greenteapress.com/matlab/spider), Spider-Man weighs 76kg.
Assume his terminal velocity is 60 m/s.
The length of the web is 100 m.
The initial angle of the web is 45° to the left of straight down.
The spring constant of the web is 40 N/m when the cord is stretched and 0 N/m when it’s compressed.