In contrast to Simon Stagg counting up to the well-known and often published escape velocity, here we have a genuine motion situation that would be familiar to any beginning student. One slight issue: the answer is wrong (both from the math as solved and from the reality of the situation we’re facing). That does not detract from the fact that this one panel is as revolutionary in its own way as the introduction of Wolverine or the birth of Phoenix.
Remember “Exact flying time withheld at request of Reed Richards!” in Fantastic Four #12? We’re going to show what the power of physics and mathematics can do with very little information.
Bruce Banner is falling eight miles towards earth through the atmosphere.
If there were no atmosphere, how long would it take and what would be his velocity at the time of impact?
Let’s look at these numbers.
First to figure out how fast Hulk will be going at impact we need to solve for the total time he’s in the air. We also need to keep our units straight. This is one of the best arguments for the metric system you will ever see.
Distance = ½ acceleration x time2
8 miles x 5280 feet / mile = ½ x 32 feet/sec2 x time2
51.4 seconds = time to Hulk’s impact.
So we should have less than a minute to Hulk’s hitting ground. If Hulk were falling through the vacuum of space to a planet with the same gravitational constant as earth’s it would be spot-on correct.
Now we can use the relationship that velocity is equal to constant acceleration multiplied by time to solve for his speed at impact time.
Velocity = acceleration x time to get our impact velocity in miles per hour
Velocity = 32 feet/second2 x 51.4 seconds x 1 mile/5280 feet x 3600 seconds/hour
Velocity = 1100 miles per hour
The text is off by a factor of 10.
The lazy …errrrr… experienced physicist would look at energy and realize that the total potential energy of an object 8 miles above the surface would be converted entirely into kinetic energy at the surface and that immediately gives the answer. Potential energy near enough to the earth is equal to mass times height times the acceleration of gravity. Kinetic energy is one-half time mass times the velocity squared. Giving us this relationship:
Mass cancels out on both sides of this equation[1] (as Galileo observed and Newton proved it should) to get velocity
= 1100 miles per hour
Two different methods of calculating the final velocity give us the same result. Huzzah!
Wait a minute you say! A heavier object does NOT fall faster than a light object? Yes. Yes I do say that. But don’t trust me or anyone else on that. Get experimental data. Consider that Women’s Single Luge and Men’s Double luge use exactly the same track. The course record speed for one woman at Lake Placid, NY is 43.985 seconds. Two men made the same distance in 43.641 seconds[2].
The Hulk’s mass dropping out of this equation also means that the biological gamma transformation he undergoes makes no difference in the calculation[3]. Bruce Banner goes from a more-or-less 80 kilogram man to a 600-plus kilogram rampaging behemoth and we do not take into account where that additional mass comes from, while being concerned about the terminal velocity of his fall is the dynamic of comic book mundane science vs. super-science in microcosm.
Now let’s turn our attention to the real-world consideration of air resistance, which any skydiver or paratrooper will tell you is significant. In general, it means that an average-sized human will reach a terminal velocity and stop accelerating. Experimental evidence shows that terminal velocity is about 200 km/hour.
Using the relationship velocity=acceleration x time as we did earlier (and keeping our units consistent), Hulk will reach terminal velocity in about 5.7 seconds, and travel 160 meters in the process, or 1.2% of his total fall of 12.9 kilometers. We’re making some simplifying assumptions and approximations here. Were Hulk a returning spacecraft or a parachuting airman we’d be working some much more intricate math.
But this leads us to a total time to landing of about 232 seconds or roughly 4 minutes, using
Colonel Joseph Kittinger (who for years had the high-altitude free-fall record) fell from 31,300 meters (almost three times Hulk’s fall) to 5,500 meters before opening his parachute. It took Kittinger roughly four minutes to go this distance (higher up he had much less air resistance) and he hit a maximum recorded speed of 988 km/hour.
Our calculated numbers for the Hulk are in rough agreement with this historic experimental result.
It is WAY more interesting to look at total energy of the Hulk, which would be mgh (mass x local gravity constant x height), assuming 600 kg for the Hulk and 9.8 m/sec2 for gravity and a height of 8 miles = 12.9 km.
In the lack of air resistance, he would land with full potential energy of (after converting units appropriately) 7.5 x 106 Joules. 1 Megajoule (1 million Joules or 106 J) is roughly equivalent to a stick of TNT — meaning the Hulk’s landing would be equivalent to about 7 sticks of TNT.
But air resistance means everything slows down. For a human this reduces the speed in the atmosphere to a final terminal velocity of roughly 200 kilometers per hour or 120 miles per hour. You can increase this speed by pointing your head down like a missile and bringing your arms in.
But figuring terminal velocity of about 200 kilometers per hour, Hulk would land with an energy of
900,000 Joules = 900,000
With air resistance the math tells us Hulk lands with an impact energy equivalent to about a stick of dynamite. This sounds about correct.
The thing to note here is that the Hulk’s potential energy at that height does not vanish – the deficit of about six sticks of dynamite goes into moving the atmosphere.
Given what we said earlier, could that energy have gone into making up the Hulk’s bulk as he transforms?
Let’s find out. The relationship of energy and mass is well-established by the Twentieth Century’s most photogenic hair-challenged theoretician:
Note that in algebra class your teacher will tell you this equation should be written with the constant as a coefficient. But some things are worth violating convention for.
So the amount of mass that 7.5 – 0.9 = 6.6 Megajoules can produce given that the speed of light is 3 x 108 meters / second is 7.3 x 10-9 kilograms. Not anywhere near enough to make up the body mass differential. The mass of a speck of dust[4] is about 7.5 x 10-10 kg, so that would be about enough energy to add ten dust specks to Bruce Banner’s mass.
Since mass and energy are directly related, let’s look at how much raw energy it would take to create the Hulk’s mass. The amount of mass converted into energy in a 20 kiloton explosion (like say, in the opening act of the Atomic Age), roughly equals 1 gram, which according to the US Bureau of Engraving and Printing is also the mass of a dollar bill. Ergo, the additional 500 Kg of mass Bruce Banner develops in becoming the Hulk would be equivalent to the energy of 500,000 Hiroshima-sized explosions. Those cool food replicators on Star Trek that make your cup of tomato soup out of pure energy? They’re marshalling the equivalent of a hydrogen bomb to put that together. Sometimes old-fashioned wet chemistry is a less expensive solution.
Jim Shooter, Editor-in-Chief at Marvel from 1978 to 1987, tells of being at a convention where Chris Claremont presented this adventure. Larry Niven, Science Fiction author and trained mathematician, was in the audience and praised the work but noted the need to take terminal velocity into account. Larry Niven confirms the event happened.
Jim Shooter’s scientific bona fides were established early on when he won a science fair contest with a project on photosynthesis including a ball and stick model of the chlorophyll molecule with marshmallows, dye, and toothpicks. Chlorophyll is not a simple molecule to model. In fact, its full stereochemistry had not been determined until a few years later in 1967. This led him into an internship in a laboratory at the University of Pittsburgh.
[1] And unlike the Anti-Life Equation – this IS an equation.
[2] From the Team USA Luge site. They have a list of all luge tracks (there are surprisingly few in the world!). https://www.teamusa.org/USA-Luge/Luge-Tracks/Lake-Placid-NY
[3] Transforming into the Hulk the cross-sectional area and aerodynamics might change – but that is a more specialized topic in aerodynamics and fluid dynamics than the relatively simple kinematics problem we’re presented with.
[4] https://www.mvorganizing.org/how-much-do-particles-weigh/