I’ve wondered for a while just how sensitive shooting is to small changes. Actually most of the possible changes in the way a person shoots are too complex to model. The angle of the right thumb and hand, for example, affect the release considerably, which affects the arrow, but that kind of thing is very difficult to calculate. There are too many variables.

So in the classic way of all physics problems, let’s imagine an idealized situation in which we’re shooting in a vacuum (no air resistance) while at the same time, the arrow spins just as it normally does in air. You have to add the second part because otherwise it’s like shooting a *makiwara-ya*, at a regular target distance… and that’s quite an adventure (don’t try it without careful safety preparations). Everything else except a few critical variables is assumed to be equal. Perhaps it will provide some idea.

So… due to gravity, the arrow will follow a parabolic arc. If you call **y** the altitude of the arrow and the angle of the arrow from horizontal at release **ϑ**, the altitude of the arrow at time **t** will be

y = V_{0}⋅sinϑ-(gt^{2}/2)

where **V _{0}** is the initial velocity of the arrow, which is to say, it’s velocity at time

**t**, the moment when it leaves the bowstring and is no longer being accelerated by the recoil of the bow.

_{0}So you can see that the altitude of the impact point depends, in this simplified case, on just the initial angle and the velocity (or just the initial angle and the time to impact, since the impact distance is fixed at 28 meters).

Unfortunately **V**_{0} depends on the *kyudoka,* the equipment, and shooting skill, so it’s highly individualized. In 「弓具の雑学事典」, published by the International Budo University, they measured **V _{x}** (the velocity along the x-axis) for ten different people with a range of experience and equipment. The initial

**V**ranged from 160-200 km/h, or more usefully, 45-56 m/s. Taking the midpoint, you get 50.5m/s which means the flight time of the arrow will be about 0.554 seconds. Since the angles are small you can assume as an approximation that

_{x}**V**=

_{0}**V**

_{x}, or just compute it:

V_{0} = V_{x}/cosϑ

Given that, you can easily figure out the height of the arrow at impact when the arrow is shot perfectly horizontally because sin0 = 0 so the first term disappears and all you have is

y_{0} = -gt^{2}/2

Given that the gravitational constant **g** = 9.80665 m/s, the point where the arrow impacts, 28 meters away, would be

y_{0} = -9.80665⋅0.554^{2}/2

y_{0} = -1.507 m

The result is negative because the arrow is falling from its initial altitude (the *kyudoka*‘s shoulder height). So here we’d expect it to fall about 151cm, which is just about right given that the average shoulder height for a Japanese man is around 150cm and of course, in reality, the arrows aren’t shot in a vacuum. The result is bound to be fuzzy due to the different arrow velocities. At the fast end of the spectrum (56m/s) it’s more like 126cm. At the slow end (45m/s) it’s 190cm. Velocity has an outsize influence because the flight time **t** is squared.

Now to figure out how much influence a small movement in the position of the arrow has, you compute the value y for a non-zero **ϑ** and compare. For reasons that will become clear, let’s use **ϑ**=0.204°. In that case the impact position becomes

y_{1} = V_{0}⋅sin(0.204)-(gt^{2}/2)

y_{1} = 50.5⋅0.00349-(9.80665⋅0.55^{2}/2)

y_{1} = -1.327

So tilting the arrow just 0.204° from horizontal at the *sha-i* (one fifth of a degree) leads to a shift in the impact position of

y_{0}-y_{1} = -1.507 – (-1.327)

y_{0}-y_{1} = -0.18

or more conveniently, 18cm… the difference between hitting the centre of the target and missing the target completely.

It’s hard to visualize one fifth of one degree, though so let’s ask, “How much of a shift in the arrow’s position corresponds to a 0.204° angle?” You can find this by asking how much the nocking point would have to shift to give 0.204° if everything else about the shot remained the same.

Call the shift along the bowstring **d** and the *yazuka* (distance from the *hazu* to the point where the arrow touches the bow at *kai*) **Z**.

sin(ϑ) = d/Z

therefore

d = Z⋅sin(ϑ)

Assuming a standard 90cm *yazuka* (the standardized value used to figure the draw weight of a *nami-sun* bow)

d = sin(0.204)*90

d = 0.321 cm

d = 3.21mm

So a shift of just three millimeters makes the difference between a bull’s eye and a miss. It seems like the same condition would apply, all else equal, if the shift were in the left hand, rather than the nocking point. So yes… pretty sensitive. This is roughly in accord with a rule of thumb I came up with a while back, that 5mm at the *sha-i* equals 18cm at 28m, so the value isn’t just theoretical, and has some experiential weight to it.

It does bring up some questions, though. One is obviously the effect of air resistance. Not only would it slow the arrow down slightly as it flies, but it would act to resist the fall of the arrow, and due to the fletchings, the arrow tip would fall more readily than the fletched end, which of course is what actually see when you shoot. Since the regulation target position is tilted back 5°, that undoubtedly represents a rule-of-thumb governing the difference in air resistance front-to-back.

Another point is the influence of arrow velocity. I never really gave it much thought before, but now the effect is more clear. Since **V**_{0} is just the result of the initial acceleration of the arrow provided by the bow and string, you start to wonder how that is engineered using different bow designs and materials (fiberglass, carbon, bamboo, and the various materials used within a bamboo bow). But I’ve never studied materials so I don’t even know what property I’m asking about. It does seem though, that for a given draw weight, some materials would be “springier” and recoil more quickly, than others. Likewise the mass and characteristics of the materials used for the arrow. Topics for another day, maybe.

And then, clearly it’s important to be precise about your nocking point, particularly as the string stretches with use. Half an arrow’s width off in either direction makes the difference between a hit and a miss.

In any case, the bottom line is that shooting is very sensitive to small changes, to the point where I can’t help thinking it’s a bit of a miracle that we can repeatedly hit the target at all, let alone 10/10. I’ve heard that some professionals, like engravers and even some competition archers, train themselves to act between heartbeats for maximum stability, but I sure haven’t. Some sort of subconscious calibration must be occurring among all the dozens, if not hundreds, of physical and mental variables involved in shooting one arrow. Fascinating stuff.

How about *enteki*? If you plug in the different parameters (58cm target at 58 meters with the target mounted 18cm above the ground) it’s not quite as sensitive: about a 9mm shift makes the difference between hitting the centre and a miss. On the other hand, as distance increases the formulas become much more sensitive to individualized values like **V**_{0} and **Z**.

Just an exercise but it’s interesting to have a rough idea of just how sensitive these things are.

That gave me ahead ache! I always hated those, what time did the train leave the station problems!

Ha! Yes, it’s a bit of a departure from my usual way of looking at things! Actually I think the real message is just how miraculous it is that we’re able to unconsciously manage all of the different factors that go into shooting, particularly when the spiritual aspect is mushin. Amazing.

Indeed, btw the seminar is over. Tomorrow the test, yosh! This year, Akiyama Sensei, kubota Sensei, Sakuma Sensei, Judges

Yosh! Indeed! Kubota-sensei taught at the International Budo Seminar in Chiba this year (something you might enjoy). I remember in particular one point where he said he’d been practising now for 50 years, putting everything into each arrow. It’s something I try to emulate, though sometimes I’m not exactly sure how!

Yes, he said 50 yrs to us also.

Thanks for the mathematical analysis! As an engineer, I was actually trying to do the same myself. I stumbled upon your calculation and I’m glad I didn’t have to reinvent the wheel 😉 That aside, your blog is great! It’s providing great insight on kyudo practice in Japan! I really have to thank you for taking the time to document what you learn. A lot of these things are orally passed down from teacher to student and never really get disseminated to others so I think this is very valuable! Of course, reading is never the same as practicing and being taught first hand by a sensei 🙂 I practice in Northern California with Seishinkan and I also just came back from the American Seminar. Best of luck to you at the All-Japan tournament if you go!

Oh, you’re welcome! It’s not my usual way of thinking about kyudo but for some reason I started thinking about it. And it is kind of good to know, and came in handy at an enteki taikai this past weekend. I hadn’t practised at that distance for months, and had new arrows, so it gave me a sense of how much to shift. Pretty good! Though now I try not to think about it too much. I hope the seminar and the testing went well for you!