Superman never made any money for saving the world from Solomon Grundy

Friday, November 14, 2008

[4-color ma] The physics of cats and (wonder) dogs

Anyone who reads genre comics - superhero in particular, but including adventure and war and western - gets used to improbable physics, those apparently impossible leaps and lifts and feats and flights. But sometimes the impossible is only unlikely.

Some time ago, Mike Sterling ran a post on Progressive Ruin entitled Adventures in Improbable Physics with Rex the Wonder Dog, in which he looked dubiously at this sequence:

Well, being a community college teacher and all, I figured it might be useful to let some students have a go at seeing if this scenario could indeed happen, at least without violating any laws of physics. So I sent the link to Dr. Burn, who is not a supervillian (although the way I render her name makes her look like one) but our resident math/physics guru. The time was right for her to introduce the problem in her class last week, and she sent me this rumination on the scenario, written while she was in her lair lab, proctoring a test:

I'm thinking about Rex the Wonder Dog. This problem is a bit complicated because of the torsional aspect of the tree branch. On one level, it is acting like a spring, which is a simple analysis. However, the spring force happens because of torque on the branch. As you know from changing a tire and using a lug wrench, torque depends on the amount of force as well as how far from the rotation point that you apply that force. The big cat has more force on the branch because it is heavier than the Wonder Dog. However, Rex is further out from the branch. So, one way to look at this is depending on the different masses, they have the same torque. But then, that wouldn't be good, because Rex would still be holding the branch and stuck in the quicksand.

But wait! The big cat "leaped" on the branch. Therefore the cat has more torque on the branch than just its own weight and a certain distance. What we can use is a conservation of energy analysis and sidestep questions about the springiness of the branch. We can say that the energy state of a closed system is always the same amount. When the big cat is at the highest point of its leap, it has maximum potential energy due being up in the air. The dog has zero potential energy because it is at ground level. The branch has zero potential energy because it hasn't been sprung yet.

At the end of the comic, the potential energy of the big cat is zero, because it is at ground level. The potential energy of the stick is zero because it isn't bent anymore by the weight of either animal (we'll ignore that it is probably still oscillating). The potential energy of the dog is at its maximum because of its height above the ground.

When we do an energy analysis, we also have to think about kinetic energy, but at the moments I am talking about, nobody is moving (at the top of a leap, you are momentarily at rest). So, kinetic energy is zero at these two points.

OK, now the numbers. Potential energy due to a spring has an equation associated with it, but we don't care because of the points we are choosing. Potential energy due to height from the ground is equal to mass*height*g where g is associated with what planet one is on (sometimes hard to determine). So, [mass of cat] *[max height it reached in its leap] is equal to [mass of dog] * [max height it reached on the ledge]. Or, height of ledge = cat to dog ratio times how high the cat leaped.
This is the scenario Dr. B was going to walk her students through. I set up an Excel spreadsheet to run some numbers and found that if we estimate the cat at 200 lbs (top end for a black panther) and Rex at 77 lbs (mid-range for a German Shepherd), and guess the cat's jump was 10 feet, the Wonder Dog could have been thrown 26 feet up to the ledge, bob's your uncle, and it's all not so improbable after all!

But wait! What about the quicksand? I emailed Dr. B to remind her of this niggling detail. She replied:

I used the comic in my class yesterday. It was a great teaching device.

I was trying to ignore the quicksand, but my students wouldn't let me. It is hard to predict the average dragging force due to the quicksand (at least for me). By the way, we can't talk about the sucking force of the quicksand, since there is no such thing as a sucking force. There is only a pressure differential. So, there could be a low pressure situation at Rex's bottom paws. Or, the frictional force could just be the scraping force as Rex gets pulled out by the branch. That is quite a range in terms of possible amount the quicksand will hinder Rex the Wonder Dog.

Frictional force is often just measured and then you know. So, we need to get a panther, the dog, the branch, and the quicksand. We can estimate how far up Rex should go. The amount under that he actually goes tells us the amount of friction. [Weight of Cat] * [height cat jumps] minus [Weight Dog]*[height dog goes]" is equal to [Friction] * [distance in the quicksand] (where distance is equal to the length of Rex's body).
Well, that didn't give me as clear an answer as I wanted. I checked some internet sources on quicksand and found this from an entry on HowStuffWorks: "Quicksand is basically just ordinary sand that has been so saturated with water that the friction between sand particles is reduced." The reason things get stuck in quicksand is that it can't support any weight because it is saturated, not because there is any sucking - a pressure differential doesn't play into it. So, the actual friction of quicksand should not be terribly high and certainly less than being stuck in a pile of regular sand.

So, if we just make the cat a little heavier, and his leap just a little higher, and the ledge just a foot or two lower, we should be able to offset any dragging force from the quicksand, and it is all still plausible.

I pity the fool who doubts the Wonder Dog.


RAB said...

Yeahhhhhhh no.

What direction is the panther leaping in? And what direction is Rex flung by the branch springing upward? And in what orientation is Rex going to be as he lands?

The panther pretty much has to be leaping forward -- i.e., over the quicksand and over Rex -- because otherwise he'll be backing up along the branch before leaping off to the nearer ground at the base of the tree trunk, and you lose much of that potential energy needed to send Rex aloft. Rex's path of motion will be towards the tree and over it, not over the panther...and he is going to land on his back, not with all four legs facing the ground. He also seems to be magically rotating along the vertical axis in mid-air, so as to be facing the panther as he lands.

In sum, the rest of it is possible if implausible, but that last panel is insane. I don't doubt the Wonder Dog, I doubt the fallible humans responsible for this scene.

(word verification: disma, like the saint.)

Scotty said...

Good points, but with one element yet unaccounted for. I think that indeed, the Panther jumped over Rex, Rex was flung away from the Panther onto the ledge. The artist perhaps struggled slightly to convey all of this in the tight panel, especially around all of the words he had to squeeze in.

But the remaining consideration is that Rex isn't a sack of potatoes or a dead weight. He's a damn Wonder Dog, and no doubt possesses a healthy amount of acrobatic skills and knowledge. It isn't difficult to rotate in the air. I imagine that between frames 2 and 3a nice front tuck layout would be just the thing.

Scotty said...

This makes me wonder, Walaka, does the venerable "180 degrees rule" ever get used in comic books? I've been thinking that if you flipped the third panel (mirror image) then the whole thing would make more sense wouldn't it? And finally, there's no reason the dog should have to land on his back (even lacking acrobatic prowess) unless he holds onto the stick for too long. I mean right? Right?!!

The Fortress Keeper said...

Physics .... schmysics! Rex the Wonder Dog transcends all human limitations.

TechWeasel said...

@Scotty - Presumably you're talking about the "axis of action" as used in film, television, and (yes) even comic books - pretty much it's often applied anytime you're trying to depict a spatial relationship graphically from a third-party viewpoint in a medium with no actual depth.

The key there is that while it's a convention it's by no means mandatory. "The Borne Ultimatum" for example is well known for ignoring axis of action in fight sequences. While it makes it nigh-impossible to follow the specifics of what exactly is going on (and gives me, personally, a headache) - it does (presumably) try to depict the disjointed relentless quality of actually being in a fight.

I think it's quite likely that the artist here is attempting the same thing, reversing center-line and axis to portray frenetic split seconds between the lion jumping, Rex's mid-air acrobatics and, ultimately, his attaining the high ground.

Clearly Rex's creative team was looking to not only push the boundaries of physics, but artistic convention as well.

Surely the wonder-dog deserves no less.

Walaka said...

Third-degree Burn sent along this additional information to illuminate the discussion further:

I simplified the problem by ignoring the direction that the panther leapt as well as the direction that Rex landed, but I didn't have to. An energy analysis consists of both potential energy (e.g. how high up you are, how much the branch is bent, how far the spring is pulled, how much TNT is in the dynamite) and kinetic energy (how fast you are moving, or spinning, or vibrating, etc.).

In an energy conservation analysis, the amount of energy originally put into the system (in this case, the system of panther, wonder dog, and branch), is the amount available at any moment. So, if we also include the panther leaping horizontally, that means there is additional energy in the system in the form of kinetic energy. Hence, Rex lands even higher.

Regarding Rex’s rotation and direction away from the branch, that would depend on when he let go of the branch. If he let go too early or too late, he would have a large horizontal velocity that would use up energy as kinetic energy. The less that goes into kinetic energy, the more available to potential energy (and therefore, more height). This quickly becomes a projectile motion problem, which is a pretty straightforward type of problem.

Rex’s rotation has to do with where his center of mass is in relationship to the force of the branch on his mouth. Since his body is hanging straight down from the branch, even as the branch moves, he won’t necessarily experience the torque that would force him to spin.

larkswindow said...

Yup, this is why I love you guys!

*shaking head*