Saturday, July 23, 2005

Falling Pennies

I was on the Royal Gorge Bridge a couple weeks ago. I heard a child ask his dad if he could drop a penny off the Bridge. The dad answered (correctly), "No."

I told a co-worker about that, and he said that he thought it was a myth that a penny dropped from up high (like the empire state building or the Royal Gorge Bridge) could kill someone, because it could never get going fast enough. Apparently he was under the impression that gravity will speed an object up to a certain velocity and then the penny would never go faster than that. I tried to explain that Gravity has a universal constant acceleration. Dropped from real high, the penny would accelerate at 9.8 meters/second2 until acted by another force... Like crashing into the head of a rafter. I told him that we could do the calculation to figure it out, but dropping it from high enough would cause the penny to go as fast as a bullet (and I did not know what height it would be).

He had it stuck in his head that there is a maximum speed a falling object can go at. Maybe he was thinking of the "acceleration of gravity". Acceleration and speed are not the same thing, but I am sure for people who don't think about it or never took physics probably could think of them as the same thing.

The fact is, of course, that acceleration is the rate at which velocity (speed) increases over time. So, if an object has a constant acceleration, by definition, it goes faster and faster. In fact, any object with a constant mass (like a penny) acted upon by a constant force (like gravity) will accelerate (go faster and faster).

You see, as Newton told us, Force = Mass * Acceleration , so Acceleration = Force / Mass

It has been calculated based on experimentation that the acceleration due to gravity is 9.8 meters/second2. This is a constant for all objects regardless of mass.

So, simple physics tells us that

Velocity = Acceleration * Time


Distance = Acceleration * Time2 /2

We happen to know the height of the bridge because I was just there and saw a sign that told us it was 1053 feet high. This is 321 meters.

Since we know the distance (321 meters) and the Acceleration (9.8 meters/second) , we can easily calculate the Time it takes for a penny to drop from the Bridge.

Time2 = 2 * Distance /Acceleration

Time = sqrt(2*321 / 9.8) = 8.09 Seconds.

Velocity = 9.8 * 8.09 = 79 meters / second

This is about 177 miles per hour.

incidentally, The average speed of a bullet on is about 950 m/s . It ranges anywhere from 180 to 1500 m/s. So, the penny is not going nearly as fast as a bullet. It would have to dropped from much higher to achieve even the slowest of the bullet speeds listed.

So, how much damage can a penny do at 79 meters / second?

Well, with some more simple physics, we can calculate the potential energy of a penny 321 meters in the air. We need its mass, but I was able to find on the internet that a penny's mass is 2.5 grams ( = .0025 kg

Energy = Force * distance = Mass * Acceleration * distance

Energy = .0025 kg * 321 meters * 9.8 meters/second2 = 7.8645 kg-meters 2 / second2 (kg-meters 2 / second2 is also known as a Joule)

So, at the point in time right before the speeding penny hits the ground, we know it has 7.8645 Joules of energy.

But how does that translate to damage?

Momentum = mass * velocity = .0025 kg * 79 meters / second = .198 kg meters / second

There is a conservation of Momentum, so if a speeding object with x Momentum crashes into an object with 0 momentum, the resulting sums of the objects' momentum after the crash will be x.

So let's assume that the penny's momentum is transferred to someone my size. I am about 108 kg.

.198 kg meters / second = 108 kg * Velocity.

My resulting Velocity could be .0018 meters / second = 1.8 mm / second.

Of course, I would not be pushed down. My head would absorbed part of the Momentum and the rest would end up in the penny bouncing off my head.

I've already spent too much time on this particular blog, so I will leave it at this:

My guess is that the Momentum contained in the falling penny would not be enough to seriously hurt at .198 kg meters / second. I looked at several "ask the physicist" sites and all agree this is not enough to seriously hurt someone.

And, of course, I am ignoring what is probably the most pertinent part of the discussion.
Air resistance on the penny, never allows the penny to get going faster than its terminal velocity. I won't do the calculations here, but if you want to see what they are, check out the site:

According to this is about
11 meters / second.

If everyone agrees that meters / second is not enough to do serious damage, we can certainly say that 11 meters / second would cause even less damage.

So I guess, in a way my co-worker was more correct than I was. Well, I guess you would have to say that he was completely correct and I was basically wrong, although I would have been correct on a planet the size of Earth with no atmosphere.

Maybe the dad should have told his son "Yes, go ahead".

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