Here's Your (co)Sine

A chicken and an egg are lying in bed. The egg is not happy. The chicken is leaning back grinning and he's puffing on a fat cigar. The egg speaks. “Well, I guess we answered that question.”

And you wonder what this has to do with shooting. (Hell, you probably wonder why you still read these blogs) As for the first question, easy, follow me here. As for the question, God only knows.

We were almost to Suttons Bay on our bicycles last summer, having left Traverse City fifteen or so miles behind us. It was uphill all the way and I was thinking of the incline. Didn't really want to think of the incline, but it was better than thinking of my sore butt which regular blog readers might remember I had sprained a number of blogs ago. (Refer to the June 10 “Call Girl” blog, if you need a refresher) The incline had reminded me of an egg joke: Why did the egg cross the road? Simple. It had the inclination. Some won’t get it. Some will. Some who do get it will wish they hadn't. Either way the joke sucks, but play along for a minute. And I promise my jokes for today about eggs are all ova...

I got to thinking about shooting up and down hills and how to compensate for bullet drop. The fact is that bullets drop based upon how far they travel horizontally, not how far they travel overall. Seems kind of counter-intuitive at first, but distance traveled vertical doesn't really matter much to the bullet.

Gravity affects a bullet basically at right angles to horizontal. So, the distance over which it affects a bullet’s flight is measured horizontally and does not count true distance on the incline. Some range finders do the math for you; they measure the true distance to the target, compute the angle up or down at which they were held, and render the horizontal distance.

If this is not taken into account, the bullet impact will be above the point of aim. How much? Well, for most of us it isn't that much. But to someone interested in the utmost precision, it will matter.

To find out how much, we need to know a few things; the actual distance to the target on the slope, the horizontal distance, the angle of the slope from horizontal, the cosine of the angle and the trajectory of the gun. If we know the measured actual distance to the target and the angle, we can calculate the horizontal distance and then plug in the rifles ballistics (and I’m sure you all know your rifle’s ballistics, right?)

(WARNING: I am about to commit the crime of math here. For you who are math professors and physicists, please forgive me! It’s been a long while since high school math class. Hell, I can’t even tell a sine from a cosine anymore… So, feel free to add your insight if you can clarify things.)

Example:

Measured distance to target on the slope=300 yards           

Angle up or down = 30 degrees

Cosine of 30 degree angle = .87 (And we all carry cosine charts, don’t we???)

So, 300 yards x .87= 261 yards.

Or, if the distance to target on the slope is 200 yards, then the actual horizontal distance is 200 x .87 which yields a horizontal distance of 174 yards.

In my 300 Blackout, which is admittedly not a flat-shooting round, the difference in the 200 yards example is two inches. If I did not compensate for the incline, I would be shooting two inches high.

The difference will be even smaller if shooting a flatter round like the 308 Win Mag. The difference here would be about .4 inches.

At a 45 degree angle, which has a cosine of .71, the actual distances from the above examples are 213 yards for 300 measured and 140 yards for the 200 yard measured.

Does this matter in Michigan? Probably not, from a practical point of view. But for those of you who head to mountainous areas, it could make the difference between a miss and a meal. And, you can impress your hunting mates around the campfire with your erudite grasp of ballistics.