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A few nights ago, Leo and I were playing with a glowing nerf bow-and-arrow-like rocket launching toy. I’m not much for toy weapons, but the “arrows” on this one are more like little rockets, they make a whizzing noise, and glow bright red, so it’s pretty fun at night in our cul-de-sac street.

Anyway, at one point Leo decided that he needed a bright white target, because it was getting quite dark, so he went inside and brought out a large sheet of art paper that he wanted me to hold up so he could shoot at it. Not really wanting to be the backstop for either his good or (more likely) poor aim, I suggested instead that it would be just as easy to lay the target on the ground.

This led to a bit of a tiff between us (the petty argument kind, not the tagged image file format kind!) Leo thought that laying it down on the street made it nearly impossible to target, whereas I thought it was more-or-less irrelevant. Later he depicted his argument with the following graphic:

That’s a bit hard to follow without the attached angry 9-year-old angrily explaining it at you, but the idea is that when the target is mounted on the wall (i.e., I’m holding it up, or it’s stuck to a tree, or something), as depicted in the center top and middle images, it is *apparently* larger than when it is laying on the ground. He made a rough guess about the ratio as the laying down apparent area being the old area divided by 2pi, or about 6x smaller.

On the left are depictions of how it’s harder to hit, in addition, because you have to shoot parabolically rather than directly at the target. I think that the the parabolic problem isn’t true — you’re always, shooting parabolically — although it may feel like you need to do that more-so because the thing is laying flat. So, I won’t pursue that part of the tiff. However, his point about the target’s apparent size is absolutely correct. Moreover, as he correctly, and angrily, pointed out, it’s worse for someone shorter!

I grasped the opportunity to redirect his ire into some trig, in order to figure out exactly how much small the target apparently is when laying down, as opposed to being mounted at eye level.

This is depicted here, and explained a little below. (I’ve slightly annotated the original page that Leo and I did the math on.)

We assumed a 6ft person (my eye level) looking at a 4ft target mounted at eye-level 20ft away. (For Leo, the observer would be about 1.5ft shorter.) We assumed as well that the target is laying flat with the center exactly at the 20ft point. We drew the two right triangles, for eye-to-far-end and eye-to-near-end of the fallen target (which would be at 22 and 18ft away along the base-line, respectively). Simple trig gave us the angles from vertical to eye-line for these, being 74.75 and 71.57 degrees, respectively. Now it’s a little geometrically tricky, but arithmetically easy to compute the length of the vertically-projected line at 20ft between the two hypotenuses (hypothenusa? hypotenua? hypotena?), but breaking it into two right triangles that are similar to the larger ones. (This can probably be done by similar triangle ratios, but it was clearer, at least at that moment, to use trig.

The result is 1.211 ft. of apparent size, which isn’t 6x smaller, but it’s about 3x smaller. And although we didn’t compute it for Leo’s height, it would obviously have been even smaller than 1.2ft apparent size. (You can prove this to yourself by observing that if all you do is to turn the paper on edge, at 6ft, where it is mounted at eye-level, it will essentially disappear altogether because you’re looking at it edge on. In general, the more parallax you have, the larger it will appear. So a shorter person with less parallax will see a small apparent target size. So it may well be that it is ~6x smaller for someone 4″9′ tall!)

Now, actually what you want is not quite this, because one’s view goes around an arc, always at 90 degrees from the center. So there’s actually a small correction required to “lean” the page slightly, as depicted by the hypotenuse on to bottom right inset. I don’t show this computation, which I did later myself, but the final size comes to about 1.14ft of apparent size (as I recall), and this is actually still slightly off since I cheated by making a right triangle with one of the sight lines, but since we’ve been wildly rounding the angles all along, a greater level of precision is definitely not called for!