There is often confusion when it comes to shooting uphill and downhill shots on how to adjust for the incline. Do you hold high on uphill shots or low on downhill shots? Vice versa? Or no change at all? The answer may surprise some people.
The key to understanding an arrow’s trajectory on uphill and downhill shots is realizing that gravity only acts perpendicular to the earth’s surface. Thus an arrow is only acted on by gravity for the distance that it travels parallel to the earth, or rather only the horizontal distance. This results in having to aim low for both downhill and uphill shots!
Because the actual distance to the target will always be greater than (or equal to) the horizontal distance to the target, an adjustment must be made when shooting at extreme angles so that the yardage used on the bow sight matches the horizontal distance. By remembering our geometry lessons, we can quickly calculate the amount of yardage that must be “cut” for either an uphill or downhill shot.
A right triangle is formed by the shooter’s height above the target, the actual distance to the target and the horizontal distance to the target. The Pythagorean theorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. By knowing the height the shooter is above the target and the actual distance to the target, the horizontal or the aiming distance can be calculated.
Rather than trying to do the calculations in your head or by hand, the following chart can be used to find the exact yardages:
| Actual Distance to Target | |||||||||||||||
| Height | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 |
| 1 | 9.9 | 15.0 | 20.0 | 25.0 | 30.0 | 35.0 | 40.0 | 45.0 | 50.0 | 55.0 | 60.0 | 65.0 | 70.0 | 75.0 | 80.0 |
| 2 | 9.8 | 14.9 | 19.9 | 24.9 | 29.9 | 34.9 | 39.9 | 45.0 | 50.0 | 55.0 | 60.0 | 65.0 | 70.0 | 75.0 | 80.0 |
| 3 | 9.5 | 14.7 | 19.8 | 24.8 | 29.8 | 34.9 | 39.9 | 44.9 | 49.9 | 54.9 | 59.9 | 64.9 | 69.9 | 74.9 | 79.9 |
| 4 | 9.2 | 14.5 | 19.6 | 24.7 | 29.7 | 34.8 | 39.8 | 44.8 | 49.8 | 54.9 | 59.9 | 64.9 | 69.9 | 74.9 | 79.9 |
| 5 | 8.7 | 14.1 | 19.4 | 24.5 | 29.6 | 34.6 | 39.7 | 44.7 | 49.7 | 54.8 | 59.8 | 64.8 | 69.8 | 74.8 | 79.8 |
| 6 | 8.0 | 13.7 | 19.1 | 24.3 | 29.4 | 34.5 | 39.5 | 44.6 | 49.6 | 54.7 | 59.7 | 64.7 | 69.7 | 74.8 | 79.8 |
| 7 | 7.1 | 13.3 | 18.7 | 24.0 | 29.2 | 34.3 | 39.4 | 44.5 | 49.5 | 54.6 | 59.6 | 64.6 | 69.6 | 74.7 | 79.7 |
| 8 | 6.0 | 12.7 | 18.3 | 23.7 | 28.9 | 34.1 | 39.2 | 44.3 | 49.4 | 54.4 | 59.5 | 64.5 | 69.5 | 74.6 | 79.6 |
| 9 | 4.4 | 12.0 | 17.9 | 23.3 | 28.6 | 33.8 | 39.0 | 44.1 | 49.2 | 54.3 | 59.3 | 64.4 | 69.4 | 74.5 | 79.5 |
| 10 | 11.2 | 17.3 | 22.9 | 28.3 | 33.5 | 38.7 | 43.9 | 49.0 | 54.1 | 59.2 | 64.2 | 69.3 | 74.3 | 79.4 | |
| 11 | 10.2 | 16.7 | 22.4 | 27.9 | 33.2 | 38.5 | 43.6 | 48.8 | 53.9 | 59.0 | 64.1 | 69.1 | 74.2 | 79.2 | |
| 12 | 9.0 | 16.0 | 21.9 | 27.5 | 32.9 | 38.2 | 43.4 | 48.5 | 53.7 | 58.8 | 63.9 | 69.0 | 74.0 | 79.1 | |
| 13 | 7.5 | 15.2 | 21.4 | 27.0 | 32.5 | 37.8 | 43.1 | 48.3 | 53.4 | 58.6 | 63.7 | 68.8 | 73.9 | 78.9 | |
| 14 | 5.4 | 14.3 | 20.7 | 26.5 | 32.1 | 37.5 | 42.8 | 48.0 | 53.2 | 58.3 | 63.5 | 68.6 | 73.7 | 78.8 | |
| 15 | 13.2 | 20.0 | 26.0 | 31.6 | 37.1 | 42.4 | 47.7 | 52.9 | 58.1 | 63.2 | 68.4 | 73.5 | 78.6 | ||
| 16 | 12.0 | 19.2 | 25.4 | 31.1 | 36.7 | 42.1 | 47.4 | 52.6 | 57.8 | 63.0 | 68.1 | 73.3 | 78.4 | ||
| 17 | 10.5 | 18.3 | 24.7 | 30.6 | 36.2 | 41.7 | 47.0 | 52.3 | 57.5 | 62.7 | 67.9 | 73.0 | 78.2 | ||
| 18 | 8.7 | 17.3 | 24.0 | 30.0 | 35.7 | 41.2 | 46.6 | 52.0 | 57.2 | 62.5 | 67.6 | 72.8 | 77.9 | ||
| 19 | 6.2 | 16.2 | 23.2 | 29.4 | 35.2 | 40.8 | 46.2 | 51.6 | 56.9 | 62.2 | 67.4 | 72.6 | 77.7 | ||
| 20 | 15.0 | 22.4 | 28.7 | 34.6 | 40.3 | 45.8 | 51.2 | 56.6 | 61.8 | 67.1 | 72.3 | 77.5 | |||
| 21 | 13.6 | 21.4 | 28.0 | 34.0 | 39.8 | 45.4 | 50.8 | 56.2 | 61.5 | 66.8 | 72.0 | 77.2 | |||
| 22 | 11.9 | 20.4 | 27.2 | 33.4 | 39.3 | 44.9 | 50.4 | 55.8 | 61.2 | 66.5 | 71.7 | 76.9 | |||
| 23 | 9.8 | 19.3 | 26.4 | 32.7 | 38.7 | 44.4 | 50.0 | 55.4 | 60.8 | 66.1 | 71.4 | 76.6 | |||
| 24 | 7.0 | 18.0 | 25.5 | 32.0 | 38.1 | 43.9 | 49.5 | 55.0 | 60.4 | 65.8 | 71.1 | 76.3 | |||
| 25 | 16.6 | 24.5 | 31.2 | 37.4 | 43.3 | 49.0 | 54.5 | 60.0 | 65.4 | 70.7 | 76.0 | ||||
| 26 | 15.0 | 23.4 | 30.4 | 36.7 | 42.7 | 48.5 | 54.1 | 59.6 | 65.0 | 70.3 | 75.7 | ||||
| 27 | 13.1 | 22.3 | 29.5 | 36.0 | 42.1 | 47.9 | 53.6 | 59.1 | 64.6 | 70.0 | 75.3 | ||||
| 28 | 10.8 | 21.0 | 28.6 | 35.2 | 41.4 | 47.3 | 53.1 | 58.7 | 64.2 | 69.6 | 74.9 | ||||
| 29 | 7.7 | 19.6 | 27.5 | 34.4 | 40.7 | 46.7 | 52.5 | 58.2 | 63.7 | 69.2 | 74.6 | ||||
| 30 | 18.0 | 26.5 | 33.5 | 40.0 | 46.1 | 52.0 | 57.7 | 63.2 | 68.7 | 74.2 | |||||
| All Measurements in Yards | |||||||||||||||
(download a .pdf of the chart here)
You may notice that there is not a lot of difference in most cases; only when the height above the target is extreme or approaching that of the horizontal distance does cutting yardage become an issue. However, the difference is there and can affect the impact site of an arrow, especially when shooting at the 12 ring on a 3D target or the vitals of a animal.
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{ 9 comments… read them below or add one }
Are the numbers in the HEIGHT column represented in feet or yards? I could not imagine being 30 YARDS above a target… Thank you…
Craig, the numbers are all in yards. While you may not find it normal to be 30 yards above a target, it is possible (though hopefully not from a treestand!) Especially when hunting out west in the mountains, steep slopes are often encountered and can present such situations. Not to mention that at many of the 3D tournaments long shots with large elevation changes are regular occurrences to really test your ability.
This sounds logical, but I have an issue. I need to drop point of aim by 6″ when shooting a target levated 6 yards and the total distance to the target is 20 yards. But If I shot on the ground I can 19.1 yards and 20 yards with no measurealbe difference. 6″ below my sight picture at 20 yards is more like 16 yards on the ground.
So, I think the angle of the gravity’s pull to the line of site must alos have an impact.
Russ,
I can fully guarantee you that the only effect that gravity has is on the horizontal path. What often happens with shooters is that they feel that there is more of an impact on elevated targets, but in reality they are usually doing something to change their shooting form and that has a larger effect on the impact position of the arrow. When shooting at elevated targets or from an elevated position, it is critical to bend at the waist and keep your anchor point identical, otherwise the vertical impact point will be affected.
I’m not saying this is the case with you, but I do know that gravity’s effect on projectiles is well known and understood.
I have seen the effect on arrow impact with downhill shots as I do most of my hunting in steep canyon lands. I shot a buck at 37 yards. My rangefinder said to shoot for 25 yards, but I thought that was a little off, so I shot for 30. The result was a high lung shot that resulted in longer suffering than should have been. He did not go far as I did not push him, but it took about 5 minutes where as a good center lung shot only takes about 30 seconds on average. I talk more about it at my website http://www.bestforhunting.com
I don’t agree with the logic of this.
Whilst the horizontal distance to the target will decrease so will the horizontal velocity of the arrow when it leaves the bow. The effect of gravity on an arrow is 4.9 t^2 metres, where t is the time of flight, assuming gravity is 9.8M/s^2 and using the simple formula S = Ut + At^2/2, where A is the accceleration due to gravity.
Assuming an arrow is fired down a slope angle Theta, range R, with speed S, then the horizontal velocity of the arrow is S cos Theta, and the horizontal range is R Cos theta, dividing them you get simple R/S as the time of flight therefore the effect of gravity on the arrow is not a function of the slope to the target. It only becomes an issue when the range to the target is measured horizontally, in which case the distance to the target is R/Cos Theta and the time of travel becomes a function of the slope. The effect is more likely due to having fletchings so as they will provide resistance to motion vertically the angle of the arrow may be affected during flight proportionally to the moment of the gravitational force acting on the arrow which will be a function of the arrows attitude.
Phil,
I’m not sure I follow your argument. However, what I can 100% with all certainty guarantee is that when shooting uphill or downhill, it’s the horizontal distance that matters. This is well know through both physics and ballistics and applies to arrows and bullets.
Gravity does have a small effect on the slowing of the arrow being shot upwards but the effect is minimal. If you were to shoot extreme distances and angles, you would find that the uphill shots would hit slightly lower than downhill shots at the same angle/distance. However, both shot scenarios would result in high hits if you were to shoot the total distance rather than the horizontal distance.
Hi Michael,
I agree that the slope does affect the height offset, but the reasoning explained here is flawed because it doesn’t take into account the reduction in the arrows horizontal velocity due to the inclination of the bow when firing at a target on a slope.
The better way to consider the problem is to rotate the reference frame so that the slope is considered horizontal.
let a slope angle be theta, and gravity G acts perpendicular to the true horizontal as we know.
considering motion along the slope in X and Y directions where X is along the slope and Y is perpendicular to the slope, for a bow speed S and angle of release Phi relative to the slope we can compute the position of the arrow at time T.
let Gx and Gy be the components of G acting on the arrow. When there is no Slope Gx = 0 and Gy = -G.
X = S.T.Cos(Phi), Y = S.T.Sin(Phi) – G.T.T/2
now assume the slope is upwards by Theta degrees and perform the same process, now Gx = -G.Sin(Theta) and Gy = -G.Cos(Theta),
X = S.T.Cos(Phi) – G.T.T.Sin(Theta)/2
Y = S.T.Sin(Phi) – G.T.T.Cos(Theta)/2
so the arrow takes longer to reach the target as it is being slowed by gravity, but it also drops less
now assume the slope is downwards by Theta degrees and perform the same process, now Gx = +G.Sin(Theta) and Gy = -G.Cos(Theta),
X = S.T.Cos(Phi) + G.T.T.Sin(Theta)/2
Y = S.T.Sin(Phi) – G.T.T.Cos(Theta)/2
so the arrow actually reaches the target faster.
I’m going to do some plots in Matlab, I may post them here later.
regards
Phil.
Hi Michael,
I created a matlab script to solve the euqations and iteratively work out the optimum offset, it’s quite surprising what the results show.
By rotating the reference plane so the slope is considered horizontal and Gravity considered to act at an angle I plotted the offset required for slopes from -45 to +45 degrees, assuming that the pin is set perfect for 100m, using a 100m target range ( along the slope ), and an arrow speed of 50 m/s, these are not realistic but show the effect.
See my comments on Red Rose Archers facebook site for plots.
interestingly it’s also very dependant on arrow speed.