There’s a great urban legend that says a penny tossed off the top of the Empire State building will impact with enough force to embed itself in the ground. Or, if it hit someone in the head, it would kill them.
Last week’s Mythbusters (one of my favorite TV shows) busted the myth experimentally. However, I wanted to understand the math and physics a little better.
The theory behind the myth looks straightforward: the Empire State Building is 1,250 feet tall. An object dropped off the top would take 8.8 seconds to reach the ground, by which time it would be moving at 193 miles an hour:
t = sqrt ( 2 * 1,250 feet / 32.2 feet/s2 ) = 8.81 seconds v = t * 32.2 feet/s2 = 283.7 feet/s = 193.4 miles per hour
This sounds impressive because we’d think of it in terms of a car — well, not my car — barreling toward something solid at unholy speeds. We’ve also seen the damage a class 5 hurricane (> 155 mph) can do. There is lots of destruction potential.
Except… there are a couple of obvious problems. First, we haven’t factored in air resistance. Air friction is one of the reasons why it is harder to bike at 20mph than it at 10mph, and why it’s harder still to do 30mph. Air resistance increases exponentially with velocity.
An object in freefall will increase its rate of descent because of gravity. For example, after one second, the object is moving at 32 feet per second. After two seconds: 64 fps. Three seconds: 96 fps. Etc. The amount of friction would progress on a geometric pattern. 1 second: 1 unit. 2 seconds: 4 units. 3 seconds: 9 units. Etc. Eventually, these “units” exceed the acceleration and the object will not go any faster.
In the magical land of physics, this is called terminal velocity, when the speed at which drag matches the pull of gravity.
Second, a penny is small. If I were to toss my Subaru off the top of the empire state building, you can bet it’s going to hurt something because my Subaru weighs about 580,000 times as much as a penny.
Mythbusters also cited the 83rd floor (which juts out and therefore catches a lot of junior scientist attempts) and the unique weather conditions generated by the large structure as myth-busting ammunition. We can all agree that either of these means the penny falls slower, so we’ll ignore them for now and look at just the terminal velocity, whose formula is:
Vterm = sqrt (
2 * m * g
c * p * A ) where:
m mass of the object
Some notes:
- The mass of pennies has changed with changes in composition. Prior to 1982, pennies weighed 3.1 grams. They got lighter when the alloy was changed from 95% copper/ 5% zinc to copper plated zinc and now weigh 2.5 grams. Terminal velocity increases proportional to the square root of mass. For our calculations, we’ll use a 1990 penny.
- Gravity accelerates a falling object at 9.81 meters per second (32.2 feet/s).
- Coefficient of drag is tricky. If a penny were molded into a sphere (which would change its area, but I’m getting ahead of myself), we’d use the value “0.5.” I’ve seen various values ranging from 0.36 to 2. Since the penny could also fall on its edge, or flat, or both, I took the average, 1.18.
- As any pilot knows, the density of the air varies with temperature, altitude, and humidity. To keep this simple, we’ll assume a temperature of 0°C, which my CRC says 1.2929 kg/m3
- The area of the object is taken from the part pointed in the direction it falls. If the penny fell flat, its area would be pi * r 2 (where r is the radius). If the penny falls on its edge, the area is 2 * r * d. What’s likely to occur is the penny will bobble around a bit. As the area falling towards the earth increases, the terminal velocity decreases. And vice versa.
- In the Mythbusters air tube test, the penny definitely did oscillate. I won’t guess how much, but we can estimate some values and come up with a good range.
Now for some data about U.S. coinage:
| units | Penny | Nickel | Dime | Quarter | |
|---|---|---|---|---|---|
| mass | kg | 0.0025 | 0.005 | 0.002268 | 0.00567 |
| thickness | m | 0.00155 | 0.00195 | 0.00135 | 0.00175 |
| diameter | m | 0.01905 | 0.02121 | 0.01791 | 0.02426 |
| Area | m^2 | 0.000285023 | 0.000353322 | 0.000251931 | 0.000462244 |
Punching up this stuff into an Excel spreadsheet, I come up with a range of 23.6 mph (penny falling flat) – to 73.3 mph (penny falling on its side — unrealistic), considerably less than the velocity in a vacuum. Mythbusters used 64mph as their upper end and fired it with a modified staple gun, showing it would do no damage (other than stinging a bit when they shot it at Adam’s posterior). They also rigged up a gun to fire pennies at near bullet speed (over ten times the terminal velocity, e.g., > 700 mph) and it still didn’t do any damage because a penny has little mass and that mass is not focused.
For grins, I’ve calculated the terminal velocities of a nickel, dime, and quarter. These hit you at 32, 24, and 28 mph, respectively. For comparisons, skydivers typically hit a terminal velocity of 120 mph. Because of their mass, they would impart a lot of damage on anything on the ground.
And for anyone who’s read this far, the ridges on the edge of the dime, quarter, half dollar and dollar were originally added as an anti-counterfeiting measure because people could subtly file off some of the (then) silver and pawn off the coin. There are 118 ridges on a dime and 119 on a quarter.