Monday morning armchair physicist

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:

  1. 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.
  2. Gravity accelerates a falling object at 9.81 meters per second (32.2 feet/s).
  3. 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.
  4. 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
  5. 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.
  6. 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.

19 Comments

  1. Very interesting. We like that show too, but we missed that episode.

  2. I really liked reading your site. Christian (a kid I babysit) and I were looking up to see if a penny could kill someone when dropped off the empire state building. I had to read him your facts about a pennies mass about 10 times for him to finally believe me that its a myth. The only thing Christian would say is, ” but, he NEVER really said that it wouldn’t kill someone.” If you have time will you send Krick an e-mail telling him it really is a myth? And it REALLY wont kill anyone> THANKS!! he would LOVE IT!

  3. If your so smart, Seriously, I really think your are, hit a golf ball 150 yards up a hill 10 ft elevation. I normally hit a 5 iron 150 yds but know that more is needed. How much? Computing the hypontenus (sp?) doesn/t work. The height of the shot, time in flight, coef. of friction, and whatever else, means that a lot more club is needed. Is this a solvableproblem? God forbid there is any wind. (I flunked algebra 2 so be too hard with the explanation)

  4. I kind of understood what you guys were saying and I kind of didn’t but it was really interesting. Could you reply back to me just telling me straight up if a penny could kill someone or not? Well thank you for your time.

    Donnie

  5. But what I really want to know is “can a racketball ball actually blind a person of normal eye health.”

  6. Thanks, I teach Jr. High Science and this was our Chapter Mystery for Motion. The site helped a lot in solving the mystery and engaging the students in understanding speed, velocity and acceleration.

  7. Wow! Talk about coincidence. I just watched the Mythbusters penny episode and thought I’d check its accuracy. Your site is terrific. Keep it going!

  8. is there a difference in the canadian nickel and the american? if so, what difference? is the nickel content more or less? the weight must be different. what do these play on the terminal velocity

  9. A Canadian nickel weighs .00395kg and is slightly thinner (0.00176 versus 0.00195 m) than an American nickel. Its composition is also different. The Canadian nickel is 94.5% steel, 3.5% copper, and 2% nickel whereas its American counterpart is 75% copper and 25% nickel.

    Plug the numbers into the spreadsheet and you’ll see the difference.

  10. There are only 118 ridges on a dime, not 188…

  11. very interesting. I’ve been trying to prove my friends wrong on this all week, and got the whole “terminal velocity” thing, but they didn’t understand it. Now I have evidence to show them. Very interesting indeed

  12. Idiots, idiots, idiots. Think. A bb, a marble, no flat side, all sides equal. Read Isaac Asimov, Physics, he has the definitive answers, though with a cannonball, round object. A round bullet shot upwards experiences wind resistance and gravity holding it back. It achieves whatever altitude possible against those two factors then fall back to earth and experiences the same two factors, gravity propelling it in reverse and wind resistence equal to that which it experienced on the way up, thus when the round bullet reaches the earth it should be travelling at muzzle velocity, same at which it left earth.
    Duh.

  13. In response to Tom-
    I’m sorry to say you are wrong, and I could go into the math, but a simple thought experiement will do- Shoot a feather or ping pong ball straight up. I can shoot it with a force (and acceleration) far greater than that of gravity, and short of destroying the object can achieve nearly any muzzle velocity I choose. On the way down however, with only the force of Earth’s gravity pulling down it against our thick atmosphere, the object will come floating back far slower than it went up. Your statement is true however on the moon.

  14. Your site is really good, it helped me understand this prac on terminal velocity i’m supposed to be doing in physics – I’m a yr 11 phys student – so, thanks very much!

    Laurel

  15. Here’s an idea. The terminal velocity of the bullet is much less than the muzzle velocity. So when the bullet comes back down, it’s not traveling at the muzzle velocity.

    Think a little harder next time, Tom.

  16. yes i saw that mythbusters episode and adam even takes it to to new levels, a penny shot to the hand and to the ass. funny as hell

  17. You have the nickel and quarter with the same mass, to three significant figures. That can’t be right. I expect the quarter to fall faster than the nickel.

    1. Author

      You’re correct about the mass – a nickel is only five grams.

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