Home Forums Hunting & Shooting Ask The Gunwriters Uphill-Downhill Trajectory Theory Question

Hello All,

My question is for any of you that know the theory, or how it can be dispelled.

With todays modern age of relatively flat-shooting calibers, what, if any, effect does slope distance play on the impact of a bullet, either uphill or downhill on a target the size of a deer?

For instance you are shooting a .308, 150gr PSP, 2820fps, sighted 2inches high at 100 yards that puts the bullet at approximately 8 inches below the line of sight at 300 yards.....
The deer being downhill at 200 yards, at a 15% grade.

The bullet hits on target at 200 yards, when shooting level.

Does the point of aim have to be adjusted?

Will a 15% grade with the target uphill be the same? If not, why?

I was always taught to aim a little low, either uphill, or downhill.....But, that distance I can not forsee enough adjustment in the point of aim to really allow for the diference as I squeeze off my shot.

Any help on this long running debate will be greatly appreciated.

Thanks,
Rockinbbar

15 degrees is about where you start seeing the effect. I was just perusing Rinker's description of this in his "firearms ballistics", and it's one of those things where even the experts argue.

The only useful thing I managed to remember from Rinker's discussion is that the most effective way to deal with angled shots is to set your sights for the horizontal distance to your target.

The distance to the target is the hypothenuse of the triangle, the rise (or fall) is the vertical, and the distance to sight for is the horizontal. The horizontal leg of the triangle is always shorter than the hypothenuse, whether shooting up or down. With angles as small as 15% and 200 yards, the difference between the actual and horizontal distance to the target is small enough that it can be neglected. HTH, Dutch.

This is a matter of physics and geometry, not theory. The trajectory still has to follow the law of gravity, and the slope (up-hill or down-hill) is still as much a geometrical angle as ever, no matter whether you're shooting a .220 Howell or a .22 Short.



Gravity pulls the bullet downward vertically from the extension of the bore axis at the moment of its exit from the muzzle, no matter what that line's angle from the horizontal. This downward pull is always vertical, always the same, no matter what the velocity, the ballistic coefficient. etc. But the amount of this drop looks different through the scope when the angle is high or low.



Many years ago, before I knew the math and science explanation for this, I figured it out for myself intuitively � which may be the best way to "explain" it for accurate but unscientific understanding and visualization �



Imagine shooting with the line of sight

(a) horizontal,

(b) vertical, muzzle-up, and

(c) vertical, muzzle-down (from a great height, as out the open bomb-bay doors of a bomber at a very high altitude).



Remember, the line of sight is at a slight angle to the bore axis.



� In situation (a), the bullet arcs slightly upward, above the line of sight a small but increasing distance, then curves over and down to cross the line of sight at the "zero" range, then drops increasingly below the line of sight.



� In situation (b), the bullet arcs away from the line of sight and curves backward overhead, farther and farther from the line of sight, never crossing the line of sight again as it reaches the peak of its upward travel and falls to earth.



� In situation (c), the bullet goes straight down along the extension of the bore axis, but the line of sight angles increasingly farther away from the extension of the bore axis, so the bullet never crosses it again.



Once you "see" these three situations clearly, you'll see that at slighter, less extreme upward and downward angles, the effect is the same, but to a lesser extent, whether the angle of the line of sight is above or below the horizontal. And the greater the angle above or below the horizontal, the farther "above" the line of sight the bullet travels on the line of flight that gravity gives it.



With a variety of muzzle velocities and ballistic coefficients, different bullets travel with slightly different divergences but always following exactly the same principles as in the extreme situations that I've just described. Trajectories of specific bullets with specific muzzle velocities and ballistic coefficients, at specific angles above or below horizontal, have to be calculated independently to get specific figures for the differences in drop below the line of sight. But the proven physical and mathematical principles that govern the outcome do not vary � ever.

Dutch and Ken are right, of course.



Gravity is what makes a bullet drop, and it doesn't care about the angle you're shooting; it'll pull the bullet towards the center of the earth. So the max drop you'll see is when your line of sight is parallel to the horizon and exactly perpendicular to gravity. You have to "aim low" on a high-sloped shot because your eyes tell you the critter is a 300 yards, say, because that's his size and proportion, but he's really only 200 yards, say, away horizontally. If you raise your muzzle to hit a 300 yard shot, you'll be too high.



In the example of a 15 degree slope at 200 yards, the difference in drop is almost insignificant. The horizontal component is 193.185 yards, so you'll see no real difference in drop in this case. A 30 degree angle would give a horizontal range of 173.205 yards, so you'd likely see some there.



Always assuming I remember this right on my old TI-68 sci calc, which is cosine (slope=15 degrees) X range in yards.



Jaywalker

Dr. Howells example is one of the better I've "seen" read on this subject, easy for me to visualize.

This is an interesting subject for me and several other folks I converse with on ocassion.

One might consider Dr. Howells example to be invalid if the rifle were sighted to be zero (0) inches high at the "zero" range (distance). This is not the case for the standard definition and method of sight mount systems used on hunting (and most other rifles). As long as the sighting system is physically (and optically) above the centerline of the bore there will be some compensation angle built into the system to allow the bore centerline to intersect the sighting system centerline. This "drop value to Zero distance" (initial angle for compensation at the zero distance) is often disreguarded/forgotten when doing trajectory correction(s) for inclined fire. The initial correction angle will typically amount to value of about 2.5 to 4 MOA for a 100 yard zero on a scoped rifle.

I don�t understand why Uphill/Downhill shooting is such a difficult concept to understand, but long ago I read an analogy (somewhere) that can hopefully make this easier to visualize.

When a scope is mounted on a rifle, there is a �fixed-relationship� created between the line-of sight (looking through the scope ... using the crosshairs to aim) and the line-of-the-bore. This relation doesn�t change ... unless you twiddle with the scopes elevation knob.

To simulate this, we�ll take a yard-stick (held out horizontally) and let its length represent a fixed distance ... (we�ll say 300 yards) along the line-of-the-bore. Crimp a small lead-shot fishing weight (representing your bullet) on a 4� length of string and attach the string to the far end of the horizontally held yardstick, so the lead-shot (your bullet) hangs down at 90 degrees. We�ll let the string represent the total drop of the bullet (from the line of the bore) at that distance. This is the drop that gravity imparts over the time (T-1) that it takes your bullet to travel 300 (horizontal) yards

Assuming your scope is mounted on (above) the yardstick, it must be angled downward to point directly at the hanging shot (your bullet). This would represent a perfect 300-yard zero.

If we now slowly elevate the yardstick to a 45 degree upward angle, this visual representation of a 300 yard shot, (with the bullet launched through your line-of bore), we can easily see what happens ... It takes the same amount of time (T-1) for your bullet to travel 300 yards, so gravity has the same effect on your bullet (ie: vertical drop is still represented by the 4� string length) .... BUT, the angle of the drop is no longer at 90 degrees to the line-of-the-bore. The lead-shot representing your bullet, is now hanging �closer� to the line-of bore ... (because gravity is working vertically). The steeper the angle that you raise up the yardstick (line-of-bore), the closer the hanging lead-shot (your bullet) comes to the line-of-the-bore (yardstick) ....until at some point (shooting nearly straight-up) ... the bullet touches the line-of bore.

You can also slowly angle the yardstick downward from the horizontal and see that the hanging lead-shot (your bullet) will similarly come closer and closer to the (extended) yardsticks line-of-the-bore .. eventually touching it, if you shoot nearly straight down.

Since the relationship between the line-of-the-bore and the scopes line-of-sight was �Fixed� when you sighted-in the rifle, measuring the bullets differing distances (drop) off the line of the bore versus the line-of sight (through the scope) is a difference in numbers only ... the relationship remains the same.

Therefore ... when shooting �steeply� up-hill/down-hill .... aim low(er).

... Silver Bullet

That�s my story ..... and I�m stickin to it !

Thanks for the detailed answers, Guys.
The question was posed elsewhere, & I was the only one that posted to adjust low, both uphill & downhill, according to distance & degree of slope.
So, I'll go back & re-define (with more info. WHY...)

Thanks again,

Rockinbbar

Put simply a bullet drops according to how far it goes horizontally. If you have a rifle sighted dead on at 200 and are aiming up or down at an animal 300yds away, but the angle up or down is steep enough that the horizontal distance is 200yds you will hit dead on.......DJ

djpaintless

Quote "If you have a rifle sighted dead on at 200 and are aiming up or down at an animal 300yds away, but the angle up or down is steep enough that the horizontal distance is 200yds you will hit dead on..."

This is not correct, it's close but not correct. You haven't considered the effect of the angle on the original 200 yard zero. For a "normal" rifle the amount of compensation necessary for a 200 yard zero would be in the 4 to 5 MOA range. Once to begin to incline the trajectory the base (200 yards in this case) zero value is also effected. The angle needed to decrease the yardage from 300 to 200 yards is ~48 degrees (Cos(48) = .67) this also effects the 4 to 5 MOA of compensation in the sighting system or the 200 yard zero. At an angle of 48 degrees only 5 (MOA) * .67 = 3.3 MOA of correction is needed, you'll still hit high by 5-3.3 = 1.7MOA or about 3.4 inches.

It ain't easy bein' senile � I plumb forgot to include the fourth imagined extreme situation that helped to make the matter clear for me all those many years ago �

� with the line of sight horizontal and the rifle up-side-down, the bullet falls farther and farther from the line of sight without ever crossing it. To a similarly head-down shooter, any impact along that bullet's trajectory would seem "high."

Don't mean to sound stupid here, but at various times throughout my life I thought I understood this subject only to later be confused by what seemed like a contradictory explanation. Could you guys break this down to "layman's" terms for those of us who have forgotten much of our high school "triggernometry" <img src="/ubbthreads/images/graemlins/wink.gif" alt="" /> and geometry.

I'll use round numbers that fit the Pythagorean theorem and go for a 3,4,5 triangle. Say the target is on top of, or at the bottom of a 300 yard tall hill, the distance from the gun to the target is 500 yards, but the actual horizonal distance is only 400 yards. (Or you could make it in feet if you think that's too far to be shooting <img src="/ubbthreads/images/graemlins/wink.gif" alt="" /> )

My understanding is that whether you drop a bullet or shoot a bullet perfectly level, assuming both are starting the same height from the ground, they would both hit the ground at the exact same time because gravity is pulling them down at the same rate. Therefore it seems to me that if I'm shooting the bullet uphill or down to a target 500 yards away, it would take a longer period of time for the bullet to get there than the horizonal distance of 400 yards and because of that extra time, the bullet would drop farther (the same distance as if it travelled 500 yards horizonally). If I'm understanding the consensus right, the horizonal distance of 400 yards is the distance to aim for? Where am I going wrong in my thinking?

What you're overlooking is the fact that when you're shooting up-hill at that pronounced angle, the line of sight is no longer horizontal, and the bore axis is inclined even higher than it'd be for a horizontal sight line. The bullet starts its flight at more of an upward angle than if it were fired toward a target on the same level as the shooter. What you're overlooking is this increase in the initial vertical component of the bullet's flight.

There are no less than three (3) "Methods" to do inclined fire that I am aware of; Rifleman's, Improved Rifleman's and the newest "Sierra Approach".

Here's a website with a very good article.

http://www.exteriorballistics.com/ebexplained/article1.html

The method you believe most here subscribe to (I don't use the horizontal distance method) is basic and easy. The Time Of Flight (TOF) as you mentioned IS NOT the same for the line-of-sight travel as it is for the horizontal distance and that's the begining of understanding the problem(s). The wind is in effect for the full line-of-sight distance (in your example, 500 yards) and the bullet is slowing down for the full time of flight so it experiences more drop than the horizontal distance (400 yards in your example) more often call the correct compensation. The Improved method uses the drop of the bullet at the ilne of sight distance and then corrects the bullet drop value by applying the COS(angle) thus giving a new bullet drop value to the target (not a new "gravity distance").

The article can describe the methods and correction better than I can here. It's a pretty good read IMHO.

The math gets really complex here, but the practical version is with modern, scoped centerfires is to disregard the angle at ranges under 200 yards. This, of course, assumes you're sighted in pretty normally, say 2" high at 100 yards, or about dead on at 200.

If the range is much past 200 yards, and the angle gets steep enough to make a significant difference in the horizontal range, then you do have to compensate.

How much? Depends on the range and the angle. I have practiced this shooting quite bit while varminting in the mountains and breaks of Montana. The last time this practice came in handy was when hunting Cape kudu in South Africa a couple years ago. I was in the bottom of a canyon, the kudu up a very steep slope. My scope reticle indicated the range was close to 400 yards, and my .30-06 was sighted in dead-on at 200.

So I held dead-on, figuring the horizontal range wasn't much more than 200, and hit just about where I aimed. The kudu dropped. Normally at that range a 180-grain '06 bullet would be down a couple feet at 400 yards.

The bullet had drifted about 4 inches, however, due to a breeze that couldn't be felt in the canyon, but could when we started climbing the mountain to retrieve the bull.

Aside from shooting myself, I've guided some big game hunters here in Montana. Most hunters overcompensate for uphill and downhill shots. With a modern rifle/cartridge combination (and here I include the .30-06, despite its age), it usually isn't something to worry about until ranges approach 300 yards and angles are pretty darn steep.

That doesn't very technical, but it comes from field shooting.

MD

The other factor is that people tend to waaaay overestimate the shot angle. Shots of more than 30 degrees are extremely rare, even in the rockies. A more typical 10 or even 20 degree angle is not going to influence anything but the very long shots.

Dave, Maybe I should have put the "Put Simply" into boldface.
If want to carry a trigonometric calculator, a bubble level, an inclinometer, and a small weather station with you to measure density altitude and temperature, you can go ahead an figure a more exact method of trajectory. Don't forget to include:
Line of Elevation
Line of Departure
The displacement between LOE and LOD
Angle of Site
Complimentary angle of Site
The sum of AOS and CAOS
Complimentary Range
Angle of Elevation
Angle of Fall
Line of Fall
Line of Impact
Angle of impact
Kinematic Viscosity
Density Altitude
Humidity
........
.......
......
.....
....
...
..
.

Or you can use that simple concept that I mentioned and that Mule Deer used to drill a Kudu 400yds away in the bottom of a valley...... <img src="/ubbthreads/images/graemlins/laugh.gif" alt="" /> <img src="/ubbthreads/images/graemlins/smile.gif" alt="" /> <img src="/ubbthreads/images/graemlins/laugh.gif" alt="" />.....DJ

It is suprising how many writers think that the bullet drop is proportional to the horizontal distance. The drop is proportional to time of flight. It is the drop that is modified by
the cosine of the angle. Ken got it right, the American Rifleman has published tables showinf it wrong.
Good luck!

Using the horizontal distance worked for me shooting steeply down into Colorado's Escalante canyon. But it may not have been a great test, as the horizontal distance was only about 200 yards, with the line of sight distance about 450, I just shot at the shoulder, hit the shoulder.