So it's not a case that Spidey's webs are probably stretchy. Rather, they have to be stretchy.

Estimating the Spring Constant of Webs

Now for some rough estimations and calculations to determine the spring constant of a web. Hold on to your pants because this might get a little wild. Let me divide this elevator's fall into three parts.

First, the elevator falls. Does it reach a state of free fall with an acceleration of -9.8m/s2? I'm not sure. Perhaps the elevator's brakes exert some drag, reducing acceleration to something less than free fall. The trailer does provide a glimpse of the brakes, so I will approximate an initial acceleration of -4.9 m/s2.

Watching the trailer, it looks like 1.5 seconds passes between the time the elevator breaks free and when Spidey's web grabs it. Assuming a constant acceleration, the elevator achieves a downward speed of 7.35 m/s. Since the elevator started from rest, I can use the average speed to find the distance traveled: 5.5 meters. That's also the unstretched length of the web (at least approximately).

Next, the elevator slows down as the web stretches. Judging from the trailer, this appears to take about 2.6 seconds. I don't know from the video if the elevator stops, but for the sake of this calculation, I will assume it does---right before a brake fails and pulls Spider-Man into the shaft. This means the elevator goes from 7.35 m/s to zero m/s in that short time. If I use the average velocity during this interval, the elevator falls 9.55 meters. Ah! But this estimation is wrong. It assumes the elevator exhibits constant acceleration during this time. It would not. As the web stretches, the force from the web increases, thereby increases the acceleration. To compensate (for now), I am going to use a stretch distance of 5 meters.

Now I can use the work energy principle to calculate the spring constant. The basic idea is to assume that both the change in kinetic energy and the change in gravitational potential energy are offset by a change in spring potential energy. The energy stored in a spring is proportional to the square of the stretch distance multiplied by the spring constant. At this point, I only need to estimate the mass of the elevator (and the human passengers) to get a value for this spring constant. Let me guess that the elevator and humans come to 1,500 kg. That gives me a spring constant of 7.76 x 104 N/m.

Does that sound crazy? Let me compare it with something to find out. What if I replace the web with a steel cable? Yes, a steel cable does indeed stretch as you pull on it. Not much, no, but it does stretch. The effective spring constant of a cable depends on three things: its length, its diameter, and its material. In general, the longer and thinner a cable, the stretchier it is. The material speaks to something called Young's modulus, and steel has a value around 200 GPa. Using this value and a cable length of 9.55 meters with a diameter of, say, 5 mm, I get a spring constant of 4 x 105 N/m. Yes, this most definitely is a higher value than the spring constant for the web, meaning it wouldn't stretch as much. But remember, stretching saves lives by decreasing the acceleration.

Homework

This is too good to not have homework questions. Ignore them if you want, but I'll leave them here to remind myself of things I can calculate later.