Incline Elevations - Long Range Shooting

Those of you who know me, know that one of my 'things' is long range shooting, ballistics and the mathematics behind it. One aspect of this is knowing how to correct your scope elevation for a particular range at different +/- inclines - for many applications (particularly at shorter ranges, ie <500 yards), the 'Riflemans Rule' or Cosine Rule is an acceptable approximation.

As an example of the Cosine Rule, consider the following: Range to Target: 500 yards; Incline: 30 degrees down; using typical ballistics for my M700 in .308" (Zero at 200 yards)

If I were shooting 'level', my elevation on the scope would be around about 8.5MOA for 500 yards. However, the inclination of -30 degrees requires an elevation correction; we will use the cosine rule to illustrate:

elevation range = linear range * cosine(inclination)

elevation range = 500 * cos (30)

elevation range = 500 * 0.866

elevation range = 433 yards

So, instead of 'dialing on' the elevation for 500 yards (8.5MOA), we 'dial on' the elevation for 433 yards - which is approx 5.5MOA. Now what does this really mean 'on target'? Well, remembering that at 500 yards, 1.0MOA is approx 5 inches and assuming that we didnt bother to correct for elevation, our 'point of impact' will be some 15 inches HIGHER than where our crosshairs are positioned! Now depending upon what you're shooting at, that is definately enough to cause a big miss!

However, the Cosine Rule is of course only an approximation (and there are better methods which will appeal to those of us who enjoy coding ballistics simulations) - the reason its only approximate is most simply summed up by saying that the Cosine Rule 'assumes' that bullet drop is perpendicular to the direct line of sight to target (ie, the 'line' on which the inclination is measured), where in fact bullet drop is always 'vertical' (ie, perpendicular to horizontal). So there are two issues here; i) the greater the inclination, the greater the error, and ii) the greater the range, the greater the error.

One a side note however, I will point out that as range increases the 'lower' your shots will drop, if you use the cosine rule - as an example, at 1000 yards and -25 deg, the Cosine Rule gives you a result of about 31MOA, where in reality your drop is in fact 34MOA (around 31 inches low!)...

So, where can we find more information on more 'precise methods'? Well luckily I have found a few interesting papers that I am currently working through, the best of these so far is this - Shooting On Slopes or, The Legend of Cosine-Range - well worth a read if this interests you.