Physics of the Everyday
# On the Field

If you’ve seen Leonardo DiCaprio’s performance in The Revenant, or wandered into a renaissance fair while lost in the park, you’ve likely encountered the beautiful art of axe throwing.

Typically, an axe is thrown so that it executes one full rotation before its head lodges into the target and a crowd of three or more onlookers genuflect in the direction of the thrower.

While it looks elegant and fluid, executing the axe throw can be quite tricky as it requires the exquisite coordination of throwing speed, spin, and aim.

The axe throw can be performed with a range of weapons, from the pocket knife to the machete to the tomahawk.

Due to the different distributions of weight, the center of mass ranges from the center of the blade for a typical throwing knife to just slightly outside the axe head for a tomahawk.

This changes the balance point during the throw.

The signature of the axe throw is the axe tumbling end over end on its way to the target.

During this complex motion, how does its center of mass move?

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**For simplicity**, focus on the case of an axe like the one we saw on the last pane.

Now that we have the basics out of the way, it’s time to get to the core of the matter.

At the moment of release, it's vital to ensure that the velocity of the axe's center of mass is pointed toward the target as that’s the direction it will move once it’s let go. But even if we time things so that the axe is released when its center of mass is on target, we still have to worry about its rotation on the way there.

The principal challenge of the axe throw is to ensure that it performs an **integer number of spins** on the way to the target, so that its head arrives in the correct orientation—this requires some coordination.

- If we set its velocity too high, will it have enough time to spin on the way to the target?
- If we set its velocity too low, will it spin fast enough to finish spinning before reaching the target?

Fortunately, its speed and its rate of spin (**angular velocity**, \(\omega\)) are not independent.
To get started, let's see how the axe rotates during our throwing motion.

When we throw the axe, our hand grips it tightly so that the axe is dragged along the curve drawn out by our arm. For simplicity, we can approximate this path as a circular arc about the elbow.

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If the tangential velocity of the axe's center of mass is \(v_\textrm{tangential},\) and the distance from the elbow to the center of mass is \(\ell,\) what is the axe's angular velocity during the throwing motion?

What we really want to know is the axe's spin in the air, as that's what will determine how it rotates on the way to the target.

Assume that we release our grip on the axe without incident, e.g. we don’t flick our wrist or change its motion in some other way.

How does its spin prior to release, \(\omega_\textrm{before},\) compare to its spin after, \(\omega_\textrm{after}?\)

For an \(n\)-loop throw to work out, the distance it travels during \(n\) rotations must be equal to the distance to the target.

Here we'll focus on one loop throws, so our central question is this:

How far does the axe travel as it makes one rotation?

Knowing the axe's angular velocity, we can write down the amount of time it takes for the axe to go through one rotation, \(T = 2\pi/\omega.\)

Find an expression for the distance \(D\) using some or all of the throwing speed \(v_\textrm{tangential},\) the angular velocity \(\omega,\) the details of the axe and throwing arm \(\ell,\) and the strength of gravity \(g.\)

Our last calculation exposes a fundamental relationship that severely constrains the condition for a successful throw.

Analyze the result and determine which is the crucial factor that ensures a successful axe throw.

Based on all your work so far, make a recommendation for the proper throwing distance to perform a one loop axe throw (in \(\si{\meter}\)).

**Assume** the length of the average adult human forearm is about \(\SI{26}{\centi\meter},\) the handle length of an axe is about \(\SI{30}{\centi\meter},\) and we hold the axe at the bottom of the handle.

One of the great joys of physics is when a simple model can explain a curious pattern in nature.

Here our crude model for the connection between the axe-throwing motion, and the axe's rate of rotation revealed a fundamental constraint, the distance required for a one-loop throw is an intrinsic function of the lengths of the axe's handle and the human forearm, \(D = 2\pi \ell.\)

No matter how hard or soft we throw the axe, it will always require the same distance to go through one-loop and thus it is a quantity over which we have no control. This is why you'll often see axe throwers theatrically counting their paces from the target: it's very important!

In general, for \(n = 1,1.5, 2, \ldots\) loop throws, etc. we have the quantization condition: \[\boxed{D = 2\pi\ell n}.\]

For common axe and forearm lengths, we found a one-loop distance of approximately 12 ft, which is indeed what you find in competitions and throwing clubs.

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