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
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.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! 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.
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.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.
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).
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.