Bullet Trajectory: Shooting Downhill and Uphill
November 13, 2015
Making precise hits when shooting downhill and uphill has morphed from dark wizardry to science over the past several decades.
As laser rangefinders came into being, and ballistic apps, precision rifles and riflescopes evolved to support longer-range shooting, dedicated riflemen delved deeper into ways to predict a bullet's path. Here's a look at several of the tools of the trade, tools that, properly used, will help you eliminate guesswork when shooting uphill and downhill.
But first, let's touch lightly on the basic physics of shooting downhill and uphill just to be sure that popular myths of the past are debunked, and that the forces that work on speeding projectiles are understood.
Many moons ago, it was popular to believe that when take shooting uphill, a bullet would hit high, and when shooting downhill, a bullet would shoot low. That notion has gone the way of iron sights. In truth, gravity exerts maximum influence on a traveling bullet when its path is level, perpendicular to the earth's pull.
When a bullet's path is angled high or low, gravity exerts more effect on velocity (which doesn't change a projectile's path appreciably) and less effect on its path. As a result, trajectory suffers less earthward "bend" and the bullet flies on a straighter path. The takeaway? Bullets hit high when shooting down and up.
Another tendency among shooters, even those of considerable skill, is to overestimate angles. I've seen lifelong old-school riflemen squint up a slope and proclaim, "That's steep... Say 30 degrees," when in reality, it's less than 15 degrees.
Compensating when shooting downhill and uphill is a science, not voodoo, but you've got to have an accurate read on the angle for your compensation to be accurate. Angle can be measured many ways, ranging from scope-mounted gadgets, to smart-phone apps, to laser rangefinders that measure and display it. We'll take a closer look at various angle-estimating tools shortly.
Once you've measured the angle, you've got to calculate how much that angle will affect your bullets' point of impact.
True ballistic distance (TBR)
A very simple way of doing this is to calculate the "true ballistic range," which is the horizontal distance between you and the target. If you're good at math (which I'm not) you can run a simple calculation based on the cosine of the angle (which you've measured) and the distance (which a rangefinder provides) and work out the horizontal distance to your target. More on that process in a bit.
Once calculated, dial your scope or hold over for the horizontal distance, and assuming an accurate rifle and correct shot execution, you'll hit your target.
A much easier and more efficient way to accomplish this is to use a rangefinder provided with an internal calculator that runs the numbers for you, and spits out the TBR. With your laser rangefinder set to produce TBR, calculating your correct holdover is as simple as ranging the target, dialing or holding for the TBR rather than the actual distance to the target, and shooting.
This process works on both uphill and downhill shots (up to 65 degrees or so) and, without doubt, is the fastest way to overcome the challenge of shooting at angles.
Angle Indicators, Cosines, and Calculations
What if your rangefinder doesn't calculate and provide TBR? The first step is to equip yourself with a device that measures angle. Scope-mounted angle indicators — also termed angle cosine indicators — are popular among long-range precision riflemen. Various manufacturers offer them, and as in all aspects of equipment, you get what you pay for. Good ones range over $100.
With the angle measured, you can reference the corresponding cosine, multiply it by the distance in yards, and use the resulting number in yards as your actual shoot-to range. The accompanying charts provide the reference numbers necessary. For instance, if the shot angle is 35 degrees, the corresponding cosine number is 0.819. Multiply your range — say 700 yards for the sake of discussion — by 0.82 (cosine rounded), and you come up with 574 yards. Dial or hold with your reticle for 574 yards, and shoot. It's effective and accurate, but slow.
A simpler way of looking at this process is to consider cosine as a percentage of the actual distance to target, which it basically is. For instance, if your angle indicator reads close to 20 degrees, cosine (0.940) percentage would indicate to shoot for 94 percent of the actual distance.
Keeping it simple, let's say that your shot distance is 1,000 yards. Hold for 940 yards, or 94 percent of the actual distance, and shoot. More math is required for most distances: for example, if your rangefinder reads 862 yards and the angle 50 degrees (cosine 0.64), unless you're a math wizard you'll have to dig out the calculator. Multiply 862 yards by .64, and you get a shoot-to distance of 552 yards.
Here's a chart, courtesy of the SAAM shooting school at the FTW ranch, showing angle cosine/actual-distance percentages:
Angle/Degree to Cosine Multiplier
Degree of angle - Multiplier of Measured Yards
5 - .99 or 99%
10 - .98 or 98%
15 - .96 or 96%
20 - .94 or 94%
25 - .91 or 91%
30 - .87 or 87%
35 - .82 or 82%
40 - .77 or 77%
45 - .70 or 70%
50 - .64 or 64%
55 - .57 or 57%
60 - .50 or 50%
65 - .42 or 42%
70 - .34 or 34%
75 - .26 or 26%
80 - .17 or 17%
85 - .09 or 9%
90 - .00 or 0%
As you can see, until angles get quite acute, bullet point of impact isn't affected all that much. Inside of 300 yards, you've got to be at a really steep angle before you need to worry about it at all. Consider this: If you're taking a shot on a mule deer with a 10-inch vital zone on a ridge, at a 30-degree angle above you, a quick calculation (300 multiplied by the 30-degree cosine percentage 0.87) will show you that your shoot-to distance is 261 yards.
A common .30-06 load pushing a 165-grain Nosler Ballistic Tip bullet impacts fractionally more than three inches of difference between 260 and 300 yards. You'll be just fine holding center on vitals, even at that quite-steep 30-degree angle.
As distance increases, angle becomes far more critical. Although very long shots are rarely taken at acute angles, for the sake of argument, let's consider a 600-yard shot with that same .30-06 load, taken at the same 30-degree angle. A calculation (600 yards multiplied by 0.87) shows the shoot-to distance as 522 yards. Between 520 yards and 600 yards the point of impact differs about 25 inches—way more than acceptable with an on-target hold.
Taking the concept to drastic lengths, let's consider the sort of uphill or downhill shot occasionally encountered by mountain goat and bighorn sheep hunters, and even desert- or alpine-country mule deer hunters: Imagine the animal you're after has bedded at the base of a massive cliff for the day, and you spent the morning hiking around and getting above it. The distance is 450 yards; well within the ethical range of an accomplished rifleman with a precision rifle.
However, the angle is almost vertical, and you've got to literally hang over the cliff face to shoot downhill. After anchoring yourself securely to a nearby boulder, you drape over, heart in your throat, and get an angle measurement on the animal far below: it's 80 degrees. Your calculation (450 yards multiplied by the 80-degree cosine percentage 0.17) establishes 76 yards as the horizontal distance, or TBR.
Here's where it gets really tricky. Because you've got your rifle sighted in at 200 yards, your point of impact is typically 1.30 inches high at 75 yards, and when pointed at 80 degrees angle, gravity isn't going to pull your bullet back down to meet your line of sight at 200 yards. Nope: that bullet is going to continue to deviate away from your line of sight and will hit considerably higher than anticipated.
If your rifle was sighted to send that bullet's path in a true parallel to your line of sight, you'd be ok, but sighted in such a manner would be completely impractical in the world of typical, almost-level shots where gravity plays a major force in directing that bullet's flight path.
So what do you do? Most savvy hunters, up against time and under extreme pressure from hanging over a cliff and attempting a very difficult shot without dropping their expensive rifle, just hold a little low on the animal if they think about it at all. In most cases, it's not enough, and an unfortunate rodeo of missed shots and cusswords follow.
Situations like this are where a smart phone with a capable ballistic app makes all the difference. You can tell your phone app where your rifle is sighted, plug in the angle, and it will tell you exactly where you'll hit. In the above situation, with the rifle zeroed at 200 yards, your 450-yard point of impact on an 80-degree shot will be over 13 inches "above" your crosshairs, even without dialing up at all. Really.
Smart-phone Apps and Angles
Another way to measure when shooting downhill or uphill is using the ballistic app on your smartphone, assuming your chosen app offer such a function. If it doesn't, add another app that does. I use the $10 version of "Ballistic," which enables me to lay my iPhone on the flat top surface of my scope's elevation turret, or on the top flat of my barrel out where it doesn't taper much, and measure the angle.
Once measured, I can lock it in and tap the calculate button, and the phone will produce holdover with the effect of the angle calculated in. It's an accurate way to go, but does take time. You've got to get the target in your crosshairs, get your phone app called up and ready to read angle, position it on your rifle, lock in the reading, run the calculation and then (finally) dial or hold and take the shot.
In the obscene-angle scenario discussed above, you've got time, and you'd best take it if you're going to make the shot. Just don't drop your phone off the cliff.
Shooting at angles can be challenging if you enjoy reaching way out and hunt, compete, or recreationally shoot in broken country. Accepting the challenge and building the skills to overcome it can be incredibly rewarding. Find a method that works for you, whether it's using a rangefinder that calculates TBR, or mounting an angle/cosine indicator on your rifle and carrying the necessary charts to enable you to run accurate calculations, or using a smartphone app, and practice with it.
If you live in flat country, you'll have to travel. For those in the West, rugged public-land country is ideal. Get out and shoot at targets of opportunity, becoming proficient with your system and building the skills necessary to make shots at any angle.
At the moment of truth, whether it be on game or on targets, an understanding of shooting downhill or uphill will give you an edge that may make all the difference.