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If a penny is dropped from the Empire State Building, then its speed, without taking air resistance into consideration, is

$\sqrt{\left(32\frac{\textrm{ft}}{\textrm{s}^2}\right)(1454\textrm{ ft})}=216.3\frac{\textrm{ft}}{\textrm{s}}$, or $147.5\frac{\textrm{mi}}{\textrm{h}}$.

What would the calculation be to take into account air resistance?

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Drag and terminal velocity require a differential equation. There is no simple closed-form answer without assuming properties of the penny, air, etc. A penny is extra hard to properly model because it has different cross sections and its actual behavior in air would need to be experimentally measured since it will tumble due to unstable air flow around it. – Brandon Enright May 24 '13 at 22:13
About $5-10 m/s$ while spinning fast, considerably higher later, see – Ikiperu May 24 '13 at 23:51

As Brandon says in his comment, the terminal velocity of a penny is difficult to calculate because its passage through the air is not only turbulent but lacking the symmetry that makes approximate calculations possible for e.g. spheres.

Under these circumstances we hedge physicists resort to experiment, and this is exactly what Myth Busters did in 2003. The terminal velocity turns out to be 65 mph. This sounds fast, but because the penny is light its energy is too low to do any harm. From my memory of the programme one of the presenters actually had a penny fired at him at 65 mph to simulate the impact (now that's dedication to science :-) and while it hurt, no damage was done.

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Are you sure they did this? I'm reading the linked page and I only see them looking at damage done in analogous situations, but seemingly taking the terminal velocity as given. Also: "Visiting the Empire State Building, the likely source of the myth, they realize that updrafts and roofs of lower floors would prevent a thrown penny from reaching street level." – Kinnisal Mountain Chicken Jun 24 '13 at 9:13
I did watch the original programme, but it was a long time ago and I have to concede I'm not sure how they arrived at the figure of 65mph. – John Rennie Jun 24 '13 at 9:16 - it appears Myth Busters didn't measure the velocity. They took 65 mph as a reasonable upper limit. – John Rennie Jun 24 '13 at 11:21
So, perhaps, that makes the OP's question into: What is that calculation on that blue board at 1:05 into the video? – Kinnisal Mountain Chicken Jun 24 '13 at 11:27

To answer your question you need to find at what speed the force of gravity is counteracted by the air resistance.

$$ m g = F_{\rm air}$$

For steady flow over a blunt object the air resistance is

$$ F_{\rm air} = \frac{1}{2} \rho A C_d v^2 $$

Given some assumptions on the density of air, and coefficient of drag you can estimate the speed $v$.

If the coin is spinning things are more complicated because the frontal area $A$ and coefficient of drag changes with time and an average needs to be calculated.

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See Brandon's comment to the question. The coin motion is chaotic and the average drag coefficient can't be calculated (or at least not without a spare supercomputer). – John Rennie Jun 24 '13 at 16:14
Well you can at least create a min. max. range to bound the answer. – ja72 Jun 24 '13 at 16:27

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