Physics Puzzle: Ain’t no Mountain High Enough
Posted on August 6, 2008
Filed Under Physics Puzzles |
Being a physicist isn’t all about solving complicated differential equations or doing complicated field theory. In fact, a good measure of a physicist’s quality is his or her ability to do “back-of-the-envelope” calculations: quick estimations of quantities of interest, done using nothing more than basic physics, a scrap of paper, some ingenuity and perfect command of the fundamentals. This week’s puzzle is one that requires such calculations.
Take a look at this beautiful mountain: the Martian Olympus Mons, courtesy of Caltech:
Its height is estimated at about 26 km - that’s about 2.6 times our highest mountain here on Earth, Mauna Kea mountain (technically it’s the highest if you take into account its underwater size, making it about 10 km tall; if you don’t then the Everest has it beat, but we’re going to look at the total size for this puzzle).
Can you explain why the Martian mountain is so much higher than any mountain here on Earth, and come up with a back-of-the-envelope calculations accounting for this factor of about 2.5 in the heights?
(EDIT, 12/8/2008: A solution has been added in the comments. Avert your eyes if you wish to try this one out for yourself!)
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7 Responses to “Physics Puzzle: Ain’t no Mountain High Enough”
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Hey, love your site. I was once upon a time a physics student, and it’s been ages since I’ve put pencil to graph paper.
My quick solution was to find the ratio of gravitational force on earth/mars.
mass(earth)= 10mass(mars) and
radius(earth) = 2 radius(mars)
so if M=mass(mars) and R=radius(mars)
F(e)/F(m)=[G10Mm/4R^2]/[GMm/R^2]
=10/4
=2.5
Very rough, and it makes the assumption that the force of gravity is proportional to the maximum height of a mountain, but for the back of an envelope it seems reasonable.
I mean inversely proportional.
That’s a good guess, but it needs to be backed up by physical intuition - *why* is the maximal height proportional to the force of gravity? You’ve got to build a “mini-model” of the problem on the back of your envelope. That’s where the interesting physics lies :).
If the mountain is too tall, the pressure at its base due to its own weight would be too large for the rock to hold it up and hence the ground beneath it would give way — i.e. it would not have been able to rise so high in the first place.
Yep, but you’re still missing the link between idea & result :). Why is the height proportional to the gravitational force? Why not, say, to its cube?
Full solution: the specific latent heat of a material determines the amount of energy it must absorb before phase-transitioning. For the base of the mountain to melt, a gravitational energy equal to its latent heat must be applied.
The gravitational energy released in “sinking” the mountain a distance d is - per unit area of the base of the mountain -
D*g*h*d
(where g - local gravitational acceleration, D - density of mountain, h - height of mountain, d - height sunk)
On the other hand, the energy needed to cause a unit area (of height d) of the base to transition from solid to liquid is:
D*d*L
For melting, we must have:
D*d*L < D*g*h*d
or:
h > L/g
Thus, the maximal height goes as ~1/g. On the Earth’s surface, g = 9.8 m/sec^2, while on Mars, g is about 4 m/sec^2. The ratio (ca. 2.5) accounts quite nicely for the difference in mountain heights.
If a mountain were to grow too high, its base would actually MELT from the pressure?! Cool!