### The simple physics behind a horrible tragedy: A tape measure with the energy of a 0.45 Colt bullet

From the NYT comes this really tragic story of a man who was killed when a tape measure from a construction site fell down 50 floors and struck him on the head. My deepest condolences to his family.

The tape measure weighed a pound so it may seem strange that it led to such an irreversible and horrible fate. Sadly the man wasn't wearing a hard hat. And physics was not on his side: as we will see below, the tape measure that struck him was tantamount to a bullet.

We can use Newton's famed three equations of motion to determine the kinetic energy of the measure as it struck the unfortunate man's head. The three equations are:

v = u + at
s = ut + at^2/2
v^2 = u^2 + 2as

Here, u is the initial velocity, v is the final velocity, a is the acceleration, s is the distance covered and t is the time.

A moment's inspection reveals that out of the three equations it's most convenient to use the third one since it does not include time, a variable that's not directly apparent in the problem. It's important to convert all units to the same MKS or CGS systems to get the right answer. We use the following values:

u = 0 since the tape measure started from a stationary state.
a = the acceleration due to gravity, g = 9.8 m/s^2
s = 400 ft = 121.9 meters
m = 1 pound = 0.45 kilograms

So v^2 comes out to be 2*9.8*121.9 = 2389.24 which we will round up to 2389.

Now the kinetic energy is just mv^2/2 so we multiply this number by the mass which is 0.45 kilograms and divide by 2.

2389*0.45/2 = 537.48, which we will round up to 537 joules.

How does this number compare to the kinetic energy of other deadly projectiles, say bullets? From this Wikipedia article on muzzle energy comes a comparison chart. 500 joules is the KE of a bullet from a 0.45 Colt pistol. The same Colt that was called "the gun that won the American West" and which was the US military's standard issue firearm until the end of the 19th century.

So the tape measure that ended a life in Jersey City today had a kinetic energy that was more than the energy of a bullet from a 0.45 Colt. It was as if the man whose belt the tape measure fell from had shot the other guy at point blank range with a 0.45 Colt. I am assuming that even with a hard hat his chances of survival might have been close to zero. But possibly finite.

This simple calculation makes as good a case as we can think of for safeguarding every piece of equipment, no matter how small or large, at the top of construction sites with your life. Doing the same math for a quarter (weighing about 6 grams) gives an energy of only 7 joules, but bump up the weight to a third of a pound and the object acquires the same KE as a bullet from a 0.22LR pistol (about 160 joules). The nature of the impact would of course also depend on the material, its shape, surface area (which is tiny for a bullet) the angle at which it strikes and other factors, but that would really be quibbling over trifles (as far as safety is concerned).

There are umpteen number of things on a construction site that are hard, rigid objects and weigh at least a third of a pound; large keychains, travel mugs, small tools like screwdrivers and cell phones come to mind. Newton's equations tell us why it's worth making sure that each one of these common necessities of daily life should be watched and secured as closely as possible. And please, please wear a hard hat.

Because everything changes when you are 400 ft from the ground.

P.S. I just took a look at my copy of Halliday and Resnick's classic physics textbook and realized that a much simpler way to do this would be to calculate the potential energy at the top - mgh. QED. This is what happens when you have not been doing physics formally for a while. It's still a good way to illustrate Newton's equations though.

1. I'm going to argue that the hard hat would have made this very survivable. The hard hat would spread the impact over the whole skull, as well as provide an inch of deceleration.

1. Yes, I am still trying to think through that one. A hard hat is meant to dissipate force and a bullet is a very tiny, concentrated source of energy so that the pressure (F/a) is much more. We'll have to work through the numbers to know how much the impact will be reduced.

2. For impact, a 3.6 kg mass impacting at 5.5 m/s, force transmitted has to be reduced to 3780 N.
For penetration, a 1 kg mass impacting at 7 m/s, can't make contact with the head.

2. " that would really be quibbling over trifles" I disagree completely. Injury from a bullet is all about penetration, this is the whole point of body armour. When considering penetration cross section and structure is critical. Consider, for example, the effect a 450g water balloon on a hard hat or skull.

1. ...except that you usually don't find water balloons on top of construction sites. Yes, the shape, density and strength of materials will all matter and a bullet is definitely a more concentrated source of energy, but by "quibbling over trifles" I meant that you should probably not be thinking of these differences when deciding whether or not to watch over your equipment on the 50th floor.

2. these calculations neglect to take into account air resistance. The tape measure would not be going as fast as calculated, unless it were felling in a vacuum. The coin in particular wouldn't end up with the velocity of a 22 cal. bullet. Same reason a leaf falling from the CN Tower wouldn't hurt anymore than a leaf from my backyard tree. Terminal velocity due to air resistance happens first.

3. Yes, I deliberately ignored air resistance (as many such problems do). I am guessing it won't impact the energy that much in terms of physiological effect.

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