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In Mary Norton's The Borrowers Aloft, the borrowers are like humans in every way except size - normal adult height is six inches. One family of three is captured by a human couple and housed in an attic, where they were to live until the couple could fabricate a display case to show them off in. They happen upon an article in a newspaper about ballooning, and afterward spend all their time alone making a gas-filled balloon contraption (like a hot-air balloon) to escape in.

[O]n that first free flight up to the ceiling... Homily [had made] the fatal mistake of jumping out of the basket as soon as it touched the floor. At terrifying speed, Pod and Arrietty had shot aloft again, hitting the ceiling with a force that nearly threw them out of the basket, while Homily--in tears-- wrung her hands below them. It took a long time to descend, even with the valve wide open, and Pod was very shaken.

"You must remember, Homily," he told her gravely when, anchored once more to the musical box, the balloon was slowly deflating, "you weigh as much as a couple of Gladstone bag keys and a roll and a half of tickets. No passenger must ever attempt to leave the car or basket until the envelope is completely collapsed." He looked very serious. "We were lucky to have a ceiling. Suppose we'd been out of doors--do you know what would have happened?"

"No," whispered Homily huskily, drying her cheeks with the back of her trembling hand and giving a final sniff.

"Arrietty and me would've shot up to 20,000 feet, and that would have been the end of us..."

"Oh dear..." muttered Homily.

"At that great height," said Pod, "the gas would expand so quickly that it would burst the canopy." He stared at her accusingly. "Unless, of course, we'd had the presence of mind to open the valve and keep it open on the whole rush up. Even then, when we did begin to descend, we'd descend too quickly. We'd have to throw everything overboard--ballast, equipment, clothes, food, perhaps even one of the passengers--"

"Oh no..." gasped Homily.

"--and in spite of all this," Pod concluded, "we'd probably crash all the same!"

Here's my question: At nearly twelve times smaller in each dimension, a borrower would have roughly 1500 times less mass than a human. Is Pod correct that a fall from a great height ("20,000 feet", though they could jump out before that) would certainly be "the end" of them? I assume their internals perform similarly per mass as ours do.

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Let's assume the drag coefficient for the borrowers is the same as that for a human. The drag force depends on the velocity and the area, which will be 144 times smaller, if your assumption of " twelve times smaller in each dimension" is correct.

\begin{align} F_D &= 0.5 \rho V^2 S C_D \\ &= 0.5 \rho C_D V^2 \frac{S_{\rm human}}{144} \\ \end{align}

For terminal velocity, the weight equals the drag force. I'm assuming the gravitational acceleration remains the same from 0 to 20,000 feet. Assuming their density is the same as ours, their mass will be $(12\times 12 \times 12)^{-1} = 1728^{-1}$ times ours.

\begin{align} m g &= 0.5 \rho C_D V^2 \frac{S_{\rm human}}{144} \\ \therefore \frac{m_{\rm human}}{1728} g &= 0.5 \rho C_D V^2 \frac{S_{\rm human}}{144} \\ \therefore V^2 &= \frac{144}{1728} \frac{2~m_{\rm human}~g}{\rho C_D S_{\rm human}} \\ V^2 &= \frac{1}{12} V_{\rm human}^2 \\ \therefore V &= 0.2887 V_{\rm human} \end{align}

These rough calculations show the terminal velocity of a borrower would be about 29% that of a human (29% of 200 km/h = 58 km/hr) , which is still pretty darn fast (ask the insects that die on your windshield when you're driving on a highway).

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  • $\begingroup$ Bone does really well in compression; smaller size actually increases a creature's odds of survival. At what height must a borrower be dropped to die? $\endgroup$ – PipperChip Dec 9 '14 at 20:30
  • $\begingroup$ @PipperChip: Well, that isn't really a physics question now, is it? There have been skydivers whose parachutes failed but they still survived, so I don't know if your question can be answered. $\endgroup$ – Pranav Hosangadi Dec 9 '14 at 20:32
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    $\begingroup$ On a recent PBS Nature program I saw, they were talking about wood ducks. Wood ducks nest up in holes in trees. About 2 days after the chicks hatch, mama bird flies down the the forest floor and calls for the chicks, who leap out of the tree, falling 20+ feet onto the forest floor, whereupon the get up and go find mama bird. They weigh on the order of an ounce. The 12x scaling yields a 100 pound person. Now, ducklings have down that may increase the drag coefficient, and maybe don't hit terminal velocity in 30-odd feet. But, a real example of small creatures doing just fine. $\endgroup$ – Jon Custer Dec 9 '14 at 21:07
  • $\begingroup$ Yes, I wonder when this crosses over into biology, but the second item we would need to know in order to determine chances of survival is how does 58 kmh affect a borrower vs. a human? In other words, not only is the terminal velocity different, but at any velocity, a borrower would land differently than a human, right? $\endgroup$ – Daniel Dec 9 '14 at 22:16
  • $\begingroup$ @Daniel, correct. After you've calculated a terminal velocity at which the borrower can expect to hit the ground, it's difficult to say anything else about survival $\endgroup$ – Pranav Hosangadi Dec 9 '14 at 22:28

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