Scaling - Why Ants Bounce But Elephants Go Splat
Drop an ant and an elephant from the Eiffel Tower. The ant lands softly and survives. The elephant goes splat and ruins the day of any nearby tourists.
An ant can lift a multiple of its own weight. An elephant can lift only a fraction of its own weight.
A small TV costs $100. A TV twice as wide costs $400.
What gives? Why is the universe like this?
The answer to this puzzle is called scaling and it is a fundamental part of physics. Learn this and you will understand many things about this world.
Linear Relationships
The equation y = x is known as a linear equation. Whatever the term 'y' represents is dependent directly on whatever the term 'x' represents. 'X' could be your paycheck and 'y' could be your bank account. Double your salary and your bank account will go up twice as fast (if you are disciplined and stay away from Amazon).
A better example is if you are walking at 5 kilometres per hour. In one hour you go 5 kilometres. In two hours you go 10 kilometres. A very simple and direct relationship.
Parabolic Relationships
The equation y = x2 is known as a parabolic equation. In this case 'y' depends on 'x' squared.
A good example is the area of a square. Take a square that is x units wide. The area of that square is x times x equals x2.
Take another square that is twice as wide (2x units wide). The area of that square is 2x times 2x equals 4x2.
If x =1 then the small square will have an area of 1 x 1 = 1 and the large square will have an area of 2 x 2 = 4.
Making TVs
This is why the price of TV's that are twice as large tend to cost something like 4 times as much. In the larger TV there are 4 times as many pixels that the factory has to produce without flaw and the odds of too many of them being dead pixels increases with the area of the panel (there is more to making a TV than that but this gives you the general idea).
More material plus a greater probability of dead pixels means cost increases faster than a simple linear relationship.
Verification
I checked this conclusion on Amazon for a range of VIZIO brand TVs. The relationship holds very well when comparing the 50 inch TV to the 25 inch TV. The relationship holds less well when comparing the 32 inch TV to the 25 inch TV because you still need to produce the supporting circuit boards and power supplies which tend to scale in price differently.
The Strength of Bones
Another good example is the strength of your bones. The strength of bones increases roughly as the area of the bone which is related the square of the width of its cross-section. Make a bone twice as wide and it can withstand four times the force before breaking.
Cubic Relationships
The equation y = x3 is known as a cubic equation. Whatever the term 'y' represents is whatever the term 'x' represents cubed.
A good example is the volume of a cube. Take a cube that is x units wide. The volume of that square is x times x times x equals x3.
Take another cube that is twice as wide or 2x units wide. The area of that square is 2x times 2x times 2x equals 8x3.
If x = 1 then the small cube will have an area of 1 x1 x 1 = 1 and the large square will have an area of 2 x 2 x 2 = 8.
A better example is the volume of a sphere which is V = 4 π r3. This is a better example because later in the post we will approximate our ant and our elephant as a spheres.
Why approximate an elephant as a sphere? Because we are physicists that's why.
Y FOR U GO SPLAT?
Okay, so now we drop our ant and our elephant from the Eiffel Tower. There is a lot going on here in the trip to the ground so let's walk through each thing one at a time.
1. Air Resistance
The air resistance will scale roughly as the area of an object which is the square of its length.
2. Mass
The mass of an object scales roughly as the volume of an object. Let's assume both the ant and the elephant are spheres. Let's say the elephant is 1000 times longer than the ant. It will therefore have a mass 1000 cubed equals 1 billion more than the ant.
Checking some numbers on the internet gives a mass of an ant at between 1 and 5 milligrams and the mass of an elephant at 3000 kg. The ratio comes out to between roughly 600 million and 3 billion. Not bad for a wild guess.
3. Force of Gravity
The force pulling down on the elephant will be F = mg and since mass is the length of the object cubed the mass will be the length of the object cubed. The force pulling down on the elephant will be much larger.
4. Total Acceleration
The force upwards, air resistance, goes up as the square of an object's size but the force downwards goes up as the cube of an object size. The two will tend to cancel each other out leaving a very rough linear relationship for the actual downward acceleration. The larger you are the faster you accelerate in free fall (in air, not in vacuum).
This means that the acceleration of the elephant will be much faster than the acceleration of the ant.
The ant will tend to waft down whereas the elephant will plummet.
It's not looking good for Mr. Elephant.
5. Bone and Body Strength
The strength of the bones and other structures within the bodies of our two creatures will tend increase with the square of the size of the dimension. These bones will have to support the mass of the creature which goes up as the cube of the creature's dimensions.
So it is harder for bones and other structures to support the increase in mass as the size of an animal increases.
This is why there is an upper size limit for land creatures. The situation in water is somewhat different so we can get larger creatures like the Blue Whale.
THE END RESULT
So because mass tends to scale as the cube of a creature's size, strength tends to scale as the square of a creature's size and the rate of falling in air is faster for larger objects the impact at the end of the trip is a very different experience for an ant and an elephant.
This is why scientists need to wear sealed hazmat suits when dropping elephants from tall towers (gross).
Thank you for reading my post.
POST SOURCES
Linear Equations
Parabolic Equations
Cubic Functions
The Power Law
How Much Do Ants Weigh?
Elephants
Great post! This problem reminds me of Google interview questions. However, I would like to elaborate on the probability discussion of pixel failure. If p is the probability of all pixels working for the 25-inch TV, then 1-p^4 would be the probability of at least one pixel dead on the 50-inch TV. Therefore, the chance of at least one dead pixel actually increase by (1+p+p^2+p^3) times. However, with modern technology, p is very close to 1 and so even though this probability increases by almost 4 times, it's still negligible.
Hey, thanks for the technical clarification. The wonderful world of manufacturing.
Dropping elephants from the Eiffel Tower to measure their propensity to splat, what a wonderful day at the office.
Almost as much fun as testing their viability as super massive elephant black hole explosion events
Physics. Proving elephants can ruin your day since 1686.
HaHa!!!:)
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I really enjoyed reading this post, you kept me engrossed from the beginning to the end.
Very good post.
"The larger you are the faster you accelerate in free fall (in air, not in vacuum)."
I'm not sure you got that one right.
I could swear I've seen it demonstrated many times where a large sphere and a small sphere are dropped from a height at the same time and they both hit the ground at the same time.
Or did I imagine it?
This greatly depends on the objects that are dropped, the height the experiment is performed from and wether or not it is conducted in a vacuum. In a vacuum a feather and a bowling ball will hit at the same time whereas outside of a vacuum the results are completely different.
As an objects mass increases it has a tendency to overcome air resistance and obtain maximum velocity quicker. At least that’s how I understand it. Please correct me if I’m wrong.
Well its obviously a bit more complicated than that e.g. the shape of the object can have a huge effect on it's falling velocity in air.
Just think of the difference between a closed and an open parachute both of which have exactly the same mass.
Great question!
A feather and a bowling ball will fall at the same rate in a vacuum. This has been proven on the Moon and inside NASA's large vacuum chamber.
Hammer vs Feather - Physics on the Moon
Watch A Bowling Ball And Feather Falling In A Vacuum
In air a light ball and a heavy ball of the same size but different mass will fall at different rates. Each will have the same air resistance but the more massive ball will have more downward gravitational pull You might need some modest equipment to prove this but it is doable.
Yes well that should be fairly obvious, but two solid balls of the same material(density) but of different size(mass/volume) will fall at the same speed/velocity.
If dropped at the same time from the same height/altitude will hit the ground at the same time.
This according to Galileo.
But other scientists say it's not true according to their formulas but don't bother to test or demonstrate their hypotheses in reality. (If you know of any such demonstrations(not formulas)please give me a link.
I have seen it successfully demonstrated on video somewhere in the UK.(can' find it now)
It's amazing that in this day and age that the answer to such a simple question could still be in any way controversial. Just like the controversy as to how an aeroplane wing gives lift.
Yes we all know about the vacuum dropping thing. They never stop going on about it.
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