When I teach ballistics, I love to ask people how you should compensate your aim when firing uphill and downhill. People usually get the downhill portion right. It’s intuitive. The bullet is “going down a hill.” The idea that it picks up speed, goes further, and thus would hit higher is quite easy for most to grasp. You compensate by aiming low. Likewise, they assume that for an uphill shot, you must aim high. The reasoning is similar; “the bullet has to climb up a hill, so it must slow down more and fall more, thus you aim higher.” This is incorrect. In both the uphill and downhill cases, you must aim lower.
For both uphill and downhill firing, the reasoning that the bullet gains or loses much steam due to the altitude change is incorrect. The difference in bullet speed by firing uphill or downhill is generally negligible compared to the deceleration due to air resistance. Rather, the major factor in the changing trajectory is that the effect of gravity is no longer full. As with most things, a picture is best:
The trajectory “bends” because of gravity. When gravity is at an angle to flight, the trajectory bends less. The more the trajectory bends, the more drop you have. The less it bends, the less drop you have. That’s why shooting uphill produces less drop. If you are still struggling with this, I give you two more pictures.
In this first picture, you are firing flat. The bullet leaves at a slightly upward trajectory to compensate for drop. This is a typical trajectory.
In this second image, you are shooting the gun straight up. Again, the direction of fire is canted slightly due to the normal angular difference between your line of sight and the actual direction the barrel is pointed. The blue trajectory was right before, but is now quite wrong. The green trajectory is correct. In this extreme case, you must aim below your target to hit it.
The Rifleman’s Rule
The Rifleman’s Rule is a simple formula to help shooters make accurate angle shots in the field. It’s quite simple:
Whether shooting uphill or downhill, correct only for the horizontal distance, not the straight line distance. In layman’s terms, do the following:
- Establish a range for your target.
- Measure the angle away from horizontal.
- Correct for range * cosine(angle)
As an example, imagine I am shooting at a target 500 meters away. I am shooting down a steep hill, at a 45° angle. Instead of correcting for 500, I correct for 500*cos(45) = 353 meters. My normal 500 meter correction for my Mk12 clone would be 3.67 mils. The correction for 353 meters is 1.87 mils. It’s an easy calculation. This method is even more field-worthy if you memorize a few big cosine values. In fact, knowing that cosine(25°) = .9, cosine(45°) = .7, and cosine(60°) = .5 will get you most of the way home in angle firing.
How accurate is this method? Well, the real calculated trajectory of this 500 meter shot at 45° is 2.23 mils. The Rifleman’s Rule is about a half mil off, or 25 centimeters at 500 meters. It’s close enough for government work, but the error is significant. The Rifleman’s rule is a great tool for field use. Even better is the angle fire function of Sendit Ballistics. This feature will be available for In-App Purchase in late April. The angle fire function adds another layer of accuracy to the trajectory, even accounting for the slight change in air density as the bullet gains or loses altitude. While you wait for this feature to become available on the App Store, give Sendit Ballistics a try. It’s free out to 500 meters.