## Unarmed and Unharmed

Shoot this puppy.

I love taking quotes out of context.  Only for comedy.  Never to win an election.

In this Buster’s Cut, the Mythbusters get pretty ambitious with physics and math.  Grant even fills up a whole bus-side full of equations!  Although the most impressive part of the whole episode was Adam’s freehand full-size cowboy drawing.  But was there substance to their style?

### De-Gunning

Jamie and Adam test the old Western cliche of the good guy being able to shoot the gun out of the bad guy’s hand leaving him “unarmed and unharmed”.  They make it pretty clear that ubiquitous shrapnel will prevent the “unharmed” bit of the myth from being fulfilled.  The “unarmed” part, however, is open for debate.

The fellas saw that they could shoot a gun out of their very unscientific metal band/Velcro hand.  But what is a (surprised) human really capable of?  They test this a couple different ways.

#### Swing Away

Adam pops out a nice little aluminum Wile E. Coyote looking gun that he lets Jaime smack with a bat.  I assume they didn’t just make up the following numbers; he said that the gun was hit with a 800 gram bat swinging at 85 mph (38 m/s).  This sounds pretty fast; they don’t specify, but for simplicity, we’ll say that it was the center of mass that that hit the gun with that speed (in fact, I guarantee the situation was far more complicated).  Additionally, apparently their bullets are 15 g traveling at 613 mph (274 m/s).  Okay, I’m following so far.  Then they say they have comparable potential energies (428 vs. 416 ft-lb).

Excuse me?

Let’s assume that everyone on the show had a brain fart, and they meant kinetic energy.  Which if it’s transferred quickly enough, has the potentialto knock a gun out of someone’s hand.  So let’s calculate it ourselves:

Bat: $\frac{1}{2} mV^{2}=\frac{1}{2}(0.8\mbox{ kg})(38\mbox{ m/s})^{2}=578\mbox{ J}=426\mbox{ ft-lbf}$
Bullet: $\frac{1}{2} mV^{2}=\frac{1}{2}(0.015\mbox{ kg})(274\mbox{ m/s})^{2}=563\mbox{ J}=415\mbox{ ft-lbf}$

Great!  Those check out amazingly well with the show.  Now here’s the rub.  Both of those are reported as simply energy.  From the standpoint of hold-on-ability, it’s not the energy nearly as much as it is the power (rate of energy transfer).  Kinetic energy isn’t even the right thing to look at here.  It is plain from the high speed that pieces of the bullet are speeding away after they break on the gun, carrying kinetic energy away that’s not left with the gun.

In fact, they make the classic mistake of interchanging units, and the narrator actual says that the two scenarios resulted in equal forces.  What’s really important here, is the momentum that’s transferred, because a square hit is going to make sure the bullet surrenders all of its forward momentum (let’s pretend it doesn’t rebound backwards, but it can rebound sideways), and we never forget, momentum is always conserved.  Compare the momentum of the two objects:

Bat: $mV=(0.8\mbox{ kg})(38\mbox{ m/s})=30\mbox{ kg m/s}$
Bullet: $mV=(0.015\mbox{ kg})(274\mbox{ m/s})=4\mbox{ kg m/s}$

Um, no wonder the bat felt worse that the bullet.  This is where the concept of impulse comes into play.  It is simply the change of momentum: $I=\Delta p=mv_{2}-mv_{1}$

Of course, if you’d like to get calculus involved (and why wouldn’t you?), you can also write impulse as:

$I=\int_{t_{1}}^{t_{2}}F\:dt$

In other words, how long it takes for momentum to transfer is nearly as important as the quantity of momentum.   If the force is constant in time, the equation simplifies to $\Delta p=F\Delta t$.  So if the bullet applies a constant force over a very short time, it will take a very large force to balance the equation.  The shorter the time, the larger the force.  They intuitively knew this after their test with the string to pull the gun out of Jaimie’s hand, which they deemed did not have enough “sting”.  Basically, the momentum transfer was smeared out as too small a force over too long a time.

#### Newton’s Law

The concept of impulse dovetails nicely into my last little commentary about the gun-slinging.  Jaime builds a nifty little device to test the actual force felt when a gun in his hand is shot (because they are rightly suspicious of their bat demonstration).  It simply adds a handle that allows him to hold a gun at an such that the barrel is point in the direction from whence a bullet may come.

Jaimie claims that Newton’s 3rd law says it will produce an equivalent force as if the gun was shot at with a bullet.  Well, yes and no.

Yes, Newton’s 3rd law is something along the lines of for every action, there’s an equal and opposite reaction.  In fact, for clarity, let’s just say for every force, there’s an equal and opposite force.

No, Newton’s 3rd law does not say the force will be the same as having your gun shot at.

Why?  The answer still lies with the time over which momentum is transferred.  They will experience the same change in momentum, but that change will manifest itself as different forces being applied over different times (remember $\Delta p=F\Delta t$ ?).  The whole point of having a barrel is that a smaller force can be applied over a longer distance, preserving the integrity of the bullet.  This exactly the opposite of bullet impact, where the bullet is slowed down in a fraction of the time, resulting in the bullet’s dismemberment.

With all that being said, in both cases, the momentum is probably delivered so quickly that a shooter wouldn’t feel the difference (all of the momentum would be transferred before the hand can react).  So it may seem I am quibbling (and I am), but their justification for what they did was shaky, at best.