Simplistic Ballistics: Shooting on Inclines

Incline_Headline

This post is part of a short series I’m calling simplistic ballistics. My goal is to illustrate some basic rules of shooting without the math heavy charts and formulas. I’m looking to lay out in clear language a core rule, and also give you an understanding on why a bullet behaves the way it does. I would caution experienced engineers and dedicated precision shooters: you should already know this, and what I’m writing here is both simplified and straightforward.

Today we’re talking about shooting on an incline. When I was in retail, I sold a lot of laser rangefinders to a lot of hunters, and had regular debates about changes to a bullet’s flight path on angles. When you’re next to the Rocky Mountains, that conversation comes up a lot, and you’d be surprised how many people are unfamiliar with this core concept:

When shooting uphill or downhill, a bullet experiences less drop than a similar shot fired on a flat plain.

This is somewhat counter intuitive, so lets take a closer look by going step by step through a simplified scenario. Our shooter is up on a cliffside, while our target is down in the bottom of the valley, like so:

Incline_blank

Using a basic rangefinder, shooter’s intuition, or god-like omnipotence: we learn the distance between us and our target:

Incline_Laser

At this point instinct says we should dial our scope for 500 meters, line up our crosshair, and squeeze the trigger. That would result in a miss when the bullet flies over the target.

You see, gravity is what causes our bullet to drop. And gravity doesn’t care about that little dip in the ground. Its force doesn’t change with the rock face. It’s always pulling straight down, towards the center of the earth. Like this:

A Note on Physics Abstraction: I realize that on a macro scale Earth's gravitational pull is spherical, and that on a micro scale gravitational pull exists between all atoms, but for the purposes of this illustration we're using gravity as the consistent force that pulls things "down."

A Note on Physics Abstraction: I realize that on a macro scale Earth’s gravitational pull is spherical, and that on a micro scale gravitational pull exists between all atoms, but for the purposes of this illustration we’re using gravity as the consistent force that pulls things “down.”

Although the bullet is travelling the full 500 meter path between us and the target, it is only being affected by 410 meters worth of gravity. The drop effect on our bullet is caused by gravity, and at an angle the bullet experiences less of it than when fired horizontally.

When we’re dialing in our scope, we want to be using that 410 meter measurement in our calculations, because that’s how much drop is being applied to the round.

Incline_Compensated

Remember that this works both ways. If the target was to take a shot at our shooter, the bullet would follow the same flight path back up the hill, and experience a similar amount of gravitational drop.

Most modern rangefinders have the ability compensate for this by using an integrated inclinometer. So they will calculate the angle difference when aiming up or down, and display the practical distance to the shooter.

Incline_degrees

If this is starting to look like basic High School trigonometry to you: Congratulations! That’s exactly what it is. And if you have no memory of High School trigonometry whatsoever, don’t panic. Almost none of us remember that stuff, the point of a good rangefinder is that it will do all that math for you in the field.

If you’re looking for more detailed and technical explanations of what’s happening here, along with all the fun formulas and charts, there’s lots of resources online from great shooters like Bryan Litz and the like. Here’s a quick reading list of similar explanations that go deeper:

Exterior Ballistics: Inclined Fire
Long Range Shooting: Angle Shooting
Wikipedia’s Entry on “the Rifleman’s Rule”



Edward O

Edward is a Canadian gun owner and target shooter with a Bachelorโ€™s Degree in Journalism. Crawling over mountains with tactical gear is his idea of fun. He blogs at TV-Presspass and tweets @TV_PressPass.


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  • jcbauerca

    Shooting up and down hill is like hitting a golf ball up or down hill. For instance, hitting a golf ball up a hill at 150 yards require a club that can hit 160 yards, while hitting a golf ball down hill requires using a club that hits it 140 yards depending on the steepness of the hill. For golfers, we use a 1 or 2 extra club factor on the steepness of the hill.

    So by the same token, I guess shooting up a hill requires a higher hold over while shooing down hill requires a lower hold over? I guess there is a simple formula that can be used to make the adjustment while on the fly.

    • Russell W.

      Nope, same hold for both. If you are not dialing the sights aim low for both

      • jcbauerca

        ok, thanks.

  • DAN V.

    I can verify that this also is true when firing a tank’s main gun.

  • ruinator

    Excellent write up Ed! Keep it coming

  • jake

    Quote: “Remember that this works both ways. If the target was to take a shot at our shooter, the bullet would follow the same flight path back up the hill, and experience a similar amount of gravitational drop.”
    Bullshit.

    Do your math again. Shooting downward, the gravitational force adds to your velocity and your bullet will reach the target in shorter time, and with higher velocity, than a bullet that is shot uphill. Add to that the fact that the bullet will be affected by the full 500 meters worth of air resistance, not just the 410 meters worth of horizontal separation and shooting up hill the cross wind will have more time to “push” on the bullet.
    It’s a lot more complicated than your basic trigonometry.

    • Jack

      Introduction of the concept. Not a masters thesis. Give the guy a break.

    • The title is “Simplistic Ballistics” for a reason ๐Ÿ™‚

    • Major Tom

      Actually his stuff is right on the money as proof of concept and in practicality. Ballistics would actually work exactly as he would describe in a frictionless environment like on the Moon.

      On Earth with an atmosphere it actually the same provided you don’t have a crosswind, headwind or tailwind. Air resistance works the same going up as it does going down, it’s why things don’t accelerate forever when in free fall.

      Additionally ballistics is a mostly symmetrical thing. Firing above a certain angle of elevation will yield approximately (wind, humidity, and various other things depending) the same impact location as if you had fired from a corresponding lower angle. To put this in perspective, in the absence of wind a round firing near-vertically at +85 degrees and a round fired at +5 degrees elevation will land in roughly the same lateral distance. It’s why mortars elevate up to aim closer.

      In short, he makes a good demonstration.

      And what’s this about gravity accelerating the bullet? A fired bullet already travels faster than gravity will ever accelerate it thus it’s irrelevant. At bullet speeds the air provides enough resistance to overcome any potential acceleration due to gravity. In fact the air resistance is enough that if you fired a gun with a -90 degree elevation (straight down), the bullet will actually decelerate as it heads towards the ground. This deceleration is constant regardless of the direction gravity pulls under Earth-normal conditions. Only in due vertical (+90 degrees) firing does gravity assist the deceleration of the bullet directly rather than simply make it arc. That’s why NASA always launches their rockets in a ballistic arc to reach orbit rather than +90 degrees due vertical. It costs less energy to reach the same altitude when traveling in an arc than it does to travel straight up. A similar effect can be observed in aircraft.

      • Logic Rules

        <<>>

        I agree with you that it wouldn’t accelerate the bullet since the deceleration from the drag would vastly exceed any acceleration from gravity. I think what Jake should have said, and perhaps what he meant to say, was that the vertical or sine component of gravity would exert some small force in the direction of travel and so it would slightly reduce velocity loss. It won’t “add to the velocity” since it won’t increase the velocity, but it will result in a higher velocity (I’d guess a very small difference) that it otherwise would be at the same flat line distance.

    • gunsandrockets

      Energy added or subtracted to a bullet because of gravity is not significant in a practical context compared to other factors affecting the bullet.

      Extreme example: bullet flight time 3(!) seconds, straight up vs straight down = a difference of approx. 200 fps.

    • redleg500

      Nope, doesn’t matter if you shooting uphill or downhill. The result is the same.

  • Russell W.

    Wind will effect it for the actual distance so you need to calculate both distances and correct for both.

  • Camilo Emiliano Rosas Echeverr

    My military friends have memorized simplified trigonometric tables just for this situations. But they are in Mountain Artillery. Weird fellows.

  • Joe Schmoe

    I don’t want to go all “know-it-all”, but this doesn’t really matter as much as some people make it out.

    We did some testing in the army, we found out that at practical ranges and situations, there is little difference. Unless you are shooting from the top of the Empire State Building or a cliff in the alps, the angle is not going to be that steep and/or the distance not that significant.

    Even in the example above (35 degrees, which is really steep) and using an M24 .308, if you set the range on the scope for around 500 meters and aim center mass, then the bullet strike on a target 410 meters away would be in the upper chest area (+/- 10-20cm); big whoop. In the example, the shooter would have been shooting down from an 86 story (286 meter high) building…

    For a competition shooting this may matter, for the military it is slightly less relevant (speaking from experience here).

    Otherwise, nice write up though ๐Ÿ™‚

    • jcbauerca

      Joe, if I understand your comment above, does it mean that if you are aiming at center of mass at 500m then you’re hitting their feet or missing short? This seems to imply that in both instances the hold over is high, is that right?

      • gunsandrockets

        No, he described hitting higher by 4 to 8 inches from the point of aim, and even that is the extreme example of a 550 yard (slant range) shot taken from the top of an 86 story building!

        • Joe Schmoe

          Yup. If you set the clicks ( sorry, don’t know US terminology) on the scope for the range of 500m.

  • TDog

    So remember kids: you WILL use geometry at some point in your life.

    Math isn’t just for nerds anymore! ๐Ÿ˜€

    • Paladin

      As a machinist I use geometry on a daily basis.

      • TDog

        Thank God! I’d hate to think how your work would turn out if you didn’t! ๐Ÿ™‚

        And just for clarification, I wasn’t knocking folks who use math or the article – I was making a lighthearted jest at all the folks who try to sound cool by saying, “You’ll never use geometry (or whatever math is the subject of conversation) when you grow up.” Hope you didn’t take it the wrong way because I really did not mean to sound like a member of the current crop of I’m-ignorant-and-I’m-proud generation.

        • Paladin

          No worries.

          And yeah, going through my classes I’ve had plenty of chances to see what happens when people without a strong grasp of geometry try operating a lathe or mill, the results can be… Less than pretty.

          I just find it kinda amusing that my job is basically the perfect counter-argument to the whole “I’ll never use this after I graduate” thing. You never know where life’s gonna take you, or what you’ll end up doing.

          • TDog

            Amen to that.

  • SlippedThroughTheCracks

    So, basically, your line of sight and line of fire is C^2, and you need to figure out B. (Assuming B is the X-axis; A is the Y-axis.)

    • Paladin

      Sort of. You can use some basic trig to figure out horizontal distance, and the change in elevation, and simply dialling for the horizontal distance should give you a decent approximation, but the more precise you want to be the more complicated the math gets.

  • Cal S.

    So, in other, engineering terms, you measure the base of the triangle instead of the hypotenuse. Easy to forget, thanks!

  • smartacus

    Waitaminute!
    Why was my first reaction to think of this as scary to figure out when i’ve had to figure out the climb time and distance and fuel usage of a Cessna based on weight, angle of attack, altitude, air pressure, air temperature, wind direction, engine power, and prop RPM?

  • Jesse Redell

    While what you’re saying is correct, your diagram is incorrect. The angle shown should be measured from the horizontal plane, not the vertical

  • Chuck

    Your simplistic logic (& every other explanation of bullet drop at slopage) forgets one extremely important – and obvious to me – fact. It is true that the bullet is affected by gravity for only the 410 meters of horizontal distance. But why does everyone seem to forget that the bullet is slowing down in relation to the entire 500 meters of travel…. not the 410 meters. So by simply dialing in the drop for 410 meters you will end up shooting several inches low. A true ballistic cheat-sheet should reflect BOTH of these factors. Think about it and you’ll know that I’m correct. I didn’t take 5 years of aeronatical engineering for nothing.

  • Franco Eldorado

    Good discussion and while it might not make a difference in the real world, it does to varmint hunters and target shooters. Variables like air pressure, temperature, and altitude also matter but for the average hunter not so much. I do like the layman explication though.

  • Hankmeister3

    In the practical world there’s not a lot of difference between “shooting on inclines” as opposed to shooting across flat ground. In some cases “shooter error” itself masks the reality of less bullet drop on inclines. However, geometrically speaking, the shooter must always remember in essence they are visually aiming along the hypotenuse while the bullet is in effect traveling along the base of a triangle with respect to actual bullet drop/distance. As a result, calculate accordingly. That’s why laser rangefinders with angle compensation are so darn handy since it does it all for you when calculating visual distance vs. “gravimetric distance”.