The Physics of Superheroes: Spectacular Second Edition (10 page)

The Green Goblin had first appeared in
Amazing Spider-M an # 14
as a mysterious crime over-boss, and grew into one of Spidey’s most dangerous foes. In addition to enhanced strength and an array of technological weapons, such as a rocket-propelled glider and pumpkin bombs, the Green Goblin managed to unmask Spider-Man and learn his secret identity in the classic
Amazing Spider-Man # 39
. Knowing that Peter Parker was really Spider-Man gave the Goblin a distinct advantage in his battles. In
Amazing Spider-Man # 121,
the Goblin kidnaps Parker’s girlfriend, Gwen Stacy, and brings her to the top of the George Washington Bridge,
¹⁶
using her as bait to lure Spider-Man into battle. At one point in their fight, the Goblin knocks Gwen from the tower, causing her to fall to her apparent doom (see figs. 6 and 7).

At the last possible instant, Spider-Man manages to catch Gwen in his webbing, narrowly preventing her from plummeting into the river below. And yet, upon reeling her back up to the top of the bridge, Spider-Man is shocked to discover that Gwen is in fact dead, despite his last-second catch. “She was dead before your webbing reached her!” the Goblin taunts. “A fall from that height would kill anyone—before they struck the ground!” Apparently the Green Goblin, creator of such advanced technology as the Goblin Glider and pumpkin bombs, suffers from a basic misunderstanding of the principle of conservation of momentum.

Fig. 6.
Gwen Stacy’s fatal plunge off the top of the George Washington Bridge, as told in
Amazing Spider-Man # 121.
Note the “SNAP” near her neck in the second to last panel.

Of course, if it were true that it was “the fall” that killed poor Gwen, then the implication for the fate of all skydivers would suggest a massive conspiracy of silence on the part of the aviation industry, not to mention that this would make the attraction of bungee jumping even more inexplicable. Nevertheless, comic-book fans have long argued over whether it was indeed the fall or the webbing that killed Gwen Stacy. This question was listed as one of the great comic-book controversies (alongside whether the Hulk is stronger than Superman, and who is faster: the Flash or Superman
¹⁷
) in the January 2000 issue of
Wizard
magazine. We now turn to physics to definitively resolve the question of the true cause of the death of Gwen Stacy.

The central question we pose is: How large is the force supplied by Spider-Man’s webbing when stopping the falling Gwen Stacy?

To determine the forces that acted upon Gwen Stacy, we first need to know how fast she was falling when the webbing stopped her. In our previous discussion of the velocity required for Superman to leap a tall building in a single bound, we calculated that the necessary initial velocity v was related to the final height h (where his speed is zero) by the expression v
²
= 2gh, where g is the acceleration due to gravity. The process of falling from a height h with initial velocity v = 0, speeding up due to the constant attractive force of gravity, is the mirror image of the leaping processes that got him to the height h in the first place.

Consequently we can employ the expression v
²
= 2gh to calculate Gwen Stacy’s speed right before she is caught in Spider-Man’s webbing. Assuming that Spidey’s webbing catches her after she has fallen approximately 300 feet, Gwen’s velocity turns out to be nearly 95 mph. Again, air resistance will slow her down somewhat, but as indicated in fig. 6, she is falling in a fairly streamlined trajectory. As we are about to discuss, the danger for Gwen is not the speed but the sudden stopping she’ll face when she hits the river.

In order to change Gwen Stacy’s motion from 95 mph to zero mph, an external force is required, supplied by Spider-Man’s webbing. The larger the force, the greater will be the change in Gwen’s velocity, or rather, her deceleration. To calculate how large a force is needed in order to bring Gwen to rest before she strikes the water, we once again turn to Newton’s second law,
F
=
ma
. Recall that the acceleration is the change in velocity divided by the time during which the speed changes. Multiplying both sides of the expression
F
=
ma
by the time over which the speed decreases, we can rewrite Newton’s second law as:

The momentum of an object is defined as the product of its mass and its speed (the right-hand side of the above equation). The product of (Force) × (time) on the left-hand side of this equation is called the Impulse. This equation, therefore, tells us that in order to change the momentum of a moving object, an external force F must be applied for a given time. The larger the interval of time, the smaller the force needed to achieve the same change in momentum.

This is the principle behind the air bags in your automobile. As your car travels down the highway at a speed of, say, 60 mph, you as the driver are obviously also moving at this same speed. When your car strikes an obstacle and stops, you continue to move forward at 60 mph, for an object in motion will remain in motion unless acted upon by an external force (that outside force is coming up in an instant). In the days before seat belts and air bags, the steering column typically supplied this external force. The time your head spent in contact with the steering wheel was brief, so consequently, the force needed to bring your head to rest was large. By rapidly inflating an air bag, which is designed to deform under pressure, the time your head remains in contact with the inflated air bag increases, compared with the steering wheel, so the force needed to bring your head to rest decreases. Distributing the force over the larger surface area of the air bag also helps to reduce injuries in a sudden stop. This force is still large enough to often knock the driver unconscious, but the important point is that it is no longer lethal. The right-hand side of the above equation is determined by the crash—you were moving at 60 mph and then you came to rest because you hit something. If you wish to avoid this, you shouldn’t have hit anything. Your mass in the expression for the change in momentum does not change (this is the best-case scenario!). Thus the right-h and side of the equation is governed by the collision, and all you can do to minimize the damage is through control of the left-h and side, that is, the impulse. The product of force and time must always be the same, as the net result is the same—namely, the initial large momentum changing to zero. The longer the time in a given impulse, the smaller the force needed to bring about the change in momentum. This is also the physical justification for a boxer rolling with a punch, increasing the time of contact between his face and his opponent’s fist, so that the force his face must supply to stop the fist is lessened.

Now, Spider-Man’s webbing does have an elastic quality, which is a good thing for Gwen Stacy, but the time that is available to slow her descent is short, which is an awful thing. For a given change in momentum, the shorter the time, the greater the necessary force. For Gwen, her change in speed is 95 mph-0 mph = 95 mph, and assuming she weighs 110 pounds, her mass in the metric system is 50 kilograms. If the webbing brings her to rest in about one half of a second, then the force applied by the webbing to break her fall is 970 pounds. Hence, the webbing applies a force nearly nine times larger than Gwen’s weight of 110 pounds. Recalling that an object’s weight is simply W = mg, where g is the acceleration due to gravity, we can say that the webbing applies a force equivalent to 9 g’s in a time span of 0.5 seconds. As indicated in fig. 6, when the webbing brings Gwen to a halt, a simple sound effect drawn near her neck (the “SNAP!” heard round the comic-book world) indicates the probable outcome of such a large force applied in such a short period of time. In contrast, bungee jumpers allow a distance sufficient for the cord to extend for many seconds, in order to keep the braking force below a fatal threshold.

Traveling at such a speed, coming to rest in such a short time interval, there is no real difference between hitting the webbing and hitting the water. However, there have been recorded cases of people surviving forces greater than that experienced by Gwen Stacy. Col. John Stapp rode an experimental rocket sled in 1954 and was subjected to a force of 40 g’s during deceleration, yet lived to describe the experience as comparable to “dental extraction without anesthetic.” Of course, Col. Stapp was securely strapped into and supported by the sled in a reinforced position. More typically, suicide victims who jump from bridges die not from drowning, but rather from broken necks. Hitting a body of water at such a speed has the same effect as hitting solid ground, as the fluid’s resistance to displacement increases the faster you try to move through it (we’ll discuss this further in Chapter 4 when considering the Flash). Tragically for Gwen Stacy, and for Spider-Man, this is another example of when comic books got their physics right, and we readers were not required to suspend our disbelief, no matter how much we may have wanted to.

Spider-Man seems to have learned this physics lesson concerning impulse and change in momentum. A story in
Spider-Man Unlimited # 2,
entitled, appropriately enough, “Tests,” finds the wall-crawler adhering to the top of a skyscraper when an unfortunate window washer plummets past him. Launching himself after the falling worker, Spider-Man must solve a real-life physics problem under more pressure than you’d find in a typical final exam. As he closes the gap between the worker (due to the fact that Spider-Man pushed off from the building with a larger initial velocity than did the window washer), Spider-Man considers: “OK, I have to do this right. Can’t snag him with a web-line, or the whiplash will get him.” As shown in fig. 8, Spider-Man recognizes that his best solution is to match his speed to that of the worker and then grab hold of him when they are barely moving relative to each other. (I’m not sure how Spider-Man slows himself down to match the worker’s velocity—perhaps by dragging his feet against the side of the building?) Then Spidey shoots out a web-line, where his arm, endowed with spider-strength, is able to withstand the large impulse associated with their upcoming change of momentum.

This solution was also employed in the 2002 motion picture
Spider-Man.
When the Green Goblin drops Mary Jane Watson from a tower on the Queensboro Bridge, in a clear homage to the storyline from
Amazing Spider-Man # 121
, Spider-Man this time does not stop her rapid descent with his webbing. Rather he dives after her, and only after catching her does he employ his webbing to swing them to (relative) safety, using the same procedure as in fig. 8. One hallmark of a hero, it appears, is the ability to learn from experience.

While certainly no hero, the above arguments have also made an impression on the Green Goblin. As mentioned earlier, the January 2000 issue of
Wizard
magazine described the controversy surrounding the death of Gwen Stacy as a classic open question in comic-book fandom. This prompted my letter to the editor of
Wizard
, published a few months later, which summarized the above physics discussion. Two years later, the August 2002 issue of
Peter Parker: Spider-Man # 45
(written by Paul Jenkins and drawn by Humberto Ramos) featured a story line in which the Green Goblin demonstrated that he had also finally learned this physics lesson. In this issue the Goblin had sent a videotape of Gwen Stacy’s death to the news media in order to psychologically torment Spider-Man. Portraying himself as the reluctant hero of this tragedy, the Goblin narrates in the tape: