The Physics Behind the Wild Wobble of Brady Ellison’s Arrows

ZOOM IN ON the details of just about any Olympic event and you can find some cool physics. Today, let’s look at the arrow in archery. It seems so simple: fletching, a shaft, and a point. It’s basically a sharp stick with some feathers. But if you watch an arrow fly in slow motion, you see something cool:

Notice how the arrow wobbles? Why does it do that? Let’s start with an arrow as it is loosed. For an approximation, I can say that only the force from the bowstring is pushing on the arrow (the gravitational force doesn’t do much during this short time interval).

With a single force on the object, the arrow will accelerate. In this case, the arrow will start from rest and increase in speed until the bowstring stops pushing on it. Really, this is the whole idea behind archery. Now, to explain the wobble, let’s consider fake forces.

What is a fake force? Well, if I go back to forces and acceleration, I can describe this relationship with Newton’s Second Law (or the Momentum Principle):

This is a great model, but it only works if I measure forces and accelerations with respect to a constant velocity reference frame (called an inertial reference frame). What if I want to look at this arrow as viewed from a reference frame that accelerates with the arrow? In that case Newton’s Second Law doesn’t work unless I add a fake force. This fake force will be equal to the mass of the arrow multiplied by the acceleration of the frame—but in the opposite direction of the frame’s acceleration.

Don’t worry about these fake forces—you’ve experienced them before. Think back to the last time you rode in an elevator. You stepped inside and the door closed. You pressed the button to take you up to the 4th floor and felt it. As the elevator accelerated upward, you felt heavier, but your mass and weight didn’t change. Instead, you experienced a fake force pushing down on you due to the acceleration of the elevator. It’s not actually a force due to a real interaction, but it feels that way.

Back to the arrow. Let’s redraw the forces on the arrow including the fake force. But where to put this fake force? For now, let’s pretend like it acts at the center of mass of the arrow.

Now we have two forces of the same magnitude pushing on the arrow. OK, try this. Get a drinking straw (a straight one) or a small stick and push on the ends like this:

Notice how the stick bends? This is a result of an imperfect stick. If you had the forces exactly aligned and the stick was completely uniform, it might simply compress and not bow. It’s just how these things behave. The same thing happens to the arrow as it is being loosed.

But what about after the arrow leaves the bow? Once the bowstring stops pushing on the arrow, the arrow is no longer accelerating (there is an air drag force, but let’s ignore that for now). If the arrow is no longer accelerating, the fake force isn’t there either. So you have a bent arrow with no forces. What do you think happens? It will return to it’s straight shape—but not stay there. When it gets to the straight position it’s also moving with some speed which causes it to overshoot this equilibrium, and now it’s bent the other way. Thus begins the oscillating arrow.

If you think that’s cool, you watch this video from Destin at Smarter Every Day that looks at the Archer’s Paradox to explain how an arrow can curve around a bow yet still hit its target.

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