Fruit Flies Like a Banana

[Explanation of title here.]

Recently, my father-in-law Scott was fashioning himself some new arrows, and I was intrigued by his set of different sized heads. He told me that the weight of the head has to be matched to the length and flexibility of the shaft to give the correct amount of wobble in the arrow's flight.

You may be surprised that arrows wobble at all, given phrases like "straight as an arrow," but if you sample the wealth of slow motion videos available, you can see it quite clearly (along with some flickering florescents):

The reason for this wobble is connected to the Archer's Paradox, which asks how an arrow can fly straight, when resting it against the bow naturally tilts it to the side. A diagram helps to understand:

Wikipedia

The left image shows the arrow notched, but not yet drawn back. B is the line between the bowstring and the body of the bow, which is the direction the force will be applied. Drawing the arrow back (right image) brings it more in line with B, but it seems like the arrow must pass through the left configuration before it leaves the bow.

The answer lies in two properties of the arrow: flexibility and tip mass. Mass resists acceleration, so when the bowstring pushes at the back of the arrow, it will begin to flex at the back, rather than push the full weight of the arrow. This will produce a wave traveling through the shaft, until it reaches the tip. The metal tip is much denser than the wood or aluminum of the body, so it is more resistant to acceleration, allowing it to stay in the position in the right image above.

I was curious if I could create a simulation of an arrow in flight. The arrow would consist of a series of line segments with mass m and applied torque of

where k is the stiffness of the arrow. This is a simple angular spring force, using the relative bend from one segment to the next. To represent the pressure of the bowstring, we can put a force on the first segment pointing forward that tapers off over time. Since the arrow is at an angle, this force will apply a torque, along with moving the arrow forward. We represent the tip by giving the last segment a larger mass than the others. Here's one example of this model:

We can try a set of different values for the tip mass and the stiffness, and look at the x-position of the arrow at the end, i.e. how straight the arrow flew:

(I could not convince PyPlot to show the tick labels properly)

While we can definitely see a relationship between the mass and the stiffness, there are some flaws in this model. For one thing, that plot shows k over 3 orders of magnitude, while mass is only varying by 0.005. It's also really slow, so I may not have run out the simulations long enough to get a good measure of the accuracy. Feel free to take a look, and make improvements!