For better or for worse, physics has always been involved in warfare. Both the machinery of the human body and the character of the theater of battle exist in the physical world, and therefore an understanding of the rules of the world can prove to be a decisive advantage in battle. This isn’t just true in the modern age of nuclear weapons, satellite reconnaissance, and laser-guided bombs. It’s been true at least since Newton’s mechanics were used to calculate the trajectory of artillery fire in the 17th century.
Those classical mechanics are still used in warfare today. The job of a modern sniper entirely depends on the physics which Newton discovered. We’ll just talk about one part of it today: bullet drop.
From our experience watching movies, we sometimes think that bullets travel to their targets instantly. The trigger is pulled and the bullet hits at essentially the same time. But for long-range shots, this is pure myth. An average handgun bullet travels well under the speed of sound. Rifle bullets (especially of the kind used for sniping) can travel at speeds of 3,000 feet per second, which is pretty close to triple the speed of sound. Fast as that is, it’s still a far cry from instantaneous at long range. The longest confirmed sniper kill was by a Canadian soldier named Rob Furlong fighting in Afghanistan, who killed a Taliban fighter at a range of just over 1.5 miles. Even at 3,000 feet per second, the bullet would still take about 2 seconds to reach its target. In reality the bullet only traveled at that speed as it left the rifle. Air resistance would begin slowing the bullet immediately, so in practice the time of flight would have been significantly longer. I’ve seen estimates of 4 seconds.
But just like a rock dropped from your hand, a bullet also begins falling once it’s been fired. The only difference is that it’s also moving forward at the same time. In 4 seconds the bullet would have fallen
some 256 feet. Therefore Rob would have to aim at a point 256 feet above the target he was trying to hit. How would he know that? Of course it’s impossible to do quickly the fairly difficult math of calculating the bullet flight time with air resistance in a combat situation. You’d at least need to know the specifics of the air pressure and wind, and the ballistic coefficient and initial velocity of the specific bullet being fired. It gets even more complicated because drag force is a function of velocity, and not a nice clean mathematically easy function either. Typically it’s more-or-less proportional to v2, but both the coefficient and the exponent tend to themselves be functions of velocity and time. Solving the differential equation analytically is generally impossible, but it can be done with numerical techniques. Fortunately the math has already been done by professional military scientists, and snipers are trained to use the results of those calculations which are printed on charts which they keep.
The skill required to bring off such a shot even with the help of physics calculations is astonishing. But it would be absolutely impossible without a strong understanding of physics.
11 responses so far ↓
1 Carl Brannen // May 25, 2008 at 6:40 pm
The military manual on sniping might be of assistance, but I recall that the record distances are with .50 BMG calibre rather than 7.62 nato.
2 CCPhysicist // May 26, 2008 at 9:44 am
You might not know that the ENIAC (what we now know to be the second electronic computer, after COLOSSUS) was developed to do ballistics calculations during WW II.
Solving the equations with a simple quadratic dependence on v would make a good problem for you!
3 anon1234 // May 27, 2008 at 9:03 am
Once upon a time I found some ballistics data published by Winchester and Remington and did some curve-fitting; as far as I could tell, the equation was mainly quadratic, but there was a cubic term that started to creep in at the higher velocities.
These were mainly .30-caliber hunting rounds; the military loads are probably a lot faster and mathematically more interesting.
4 CCPhysicist // May 27, 2008 at 6:10 pm
The mechanical fire control computers on battleships did not compute a trajectory, they computed a firing solution based on ballistics calculations that were only done once for each condition (windage, in particular) that was not as simple as the velocities and latitude of the ships.
Interestingly, I learned they were not replaced when the ships were brought back into service during the electronic computer era. They could fight the ship on human power alone.
Coolest fact of all: the optimum launch angle for the giant railway-based guns used in WW I was above 45 degrees. You get more range by getting up where the atmosphere is thin and there is less drag.
5 CCPhysicist // May 28, 2008 at 2:35 pm
The difference between a firing solution and a full calculation of the trajectory is the difference between what the BRL did (and built ENIAC to replace a room full of desktop Marchant calculators) and what modern anti-cannon technology does.
BRL solution:
Given a particular 16″ shell and a discrete number of bags of powder, plus some test firings on a range under varying conditions, you work out the range-angle relationship for that round under all possible conditions and then manufacture cams and out-of-round gears, etc to mechanically reproduce the functional relationship you have drawn on paper that interpolates between what you know based on what you have calculated. That is how the battleship’s analog computer can give you a firing solution in zero time. (As soon as you change the angle of the little ship showing the heading of the target, the angle and azimuth of the gun changes.) It does what you would do by looking up that case in a book of tables.
Modern solution:
Given the position and velocity of an incoming shell at several points determined by radar and an estimate of the mass and diameter of the projectile (155 mm howitzer? mortar?), actually solve the system of equations from t=T back to t=0 to locate the source, and shoot back. You use a digital computer program to solve the differential equation in real time. Or so I guess based on what I have read and know I can do.
6 jimi hendrix // Jul 11, 2008 at 9:41 pm
wat is the theory for the sniper rifles bullet trajectory is it carolus theory?
7 Matthew La Croix // Jul 21, 2008 at 12:05 am
Jimi, what you’re referring to is called the Coriolis Effect but it has more to do with the earth’s rotation in correlation to a bullet’s path to an intended target, rather than the earth’s shape (this effect is partly responsible for giving a curve-ball its “curve” as well). As a bullet leaves a sniper’s rifle, it appears to drop more with longer distances to its intended target, even though it’s really traveling in as straight a path as possible which is why snipers and artillery teams are specifically trained to take this into consideration when figuring a firing solution from great distances (…moreso with artillery rounds).
If you’ve seen the movie Wanted recently, the opening and closing sniping scenes are not improbable shots to acquire but, based quiet firmly in mathematical fact and can be achieved through the extensive use of a physics formula I’d rather not attempt to explain in this forum. The shot achieved in the movie, however would not require the use of any scope per se, but would indeed require that the intended target be placed within a particular spot at the end of the bullet’s trajectory after being fired…hence, an X would most definitely have to mark the spot. The Germans had this engineered almost to an art when they used the Kaiser Wilhelm Geschütz or “Paris-Gun” to shell the french capital from 75+ miles away during WWI…the 200+ round took a total of 170 seconds to find it’s intended target after being fired.
I hope that you found this helpful.
8 Ross Pruitt // Nov 12, 2008 at 11:08 pm
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9 Lviper // Nov 25, 2008 at 9:06 pm
coriolis does not effect precision shots to any sort of noticeable degree inside of 1600 yds at see level. The Earth drags its atmosphere with it. I figured I would put that out there as a practical assertion. With all precision “sniper” shots taken there is a math answer and a real answer. Snipers make their own books- D.O.P.E. books (data of previous engagements) to predict what bullets will do under certain conditions.
10 Lviper // Nov 25, 2008 at 9:10 pm
btw the canadian shot was 1700 with a .300 winmag and the shot was made at upwards of 4ooo feet. the adjustment made was for a shot that at sea level would be equivalent of an 800 yard shot. which is well inside of the effective range using 190 grain bullets (1250 by the book). excellent considering he made the adjustment with density altitude at the top of his list.
11 Lviper // Nov 25, 2008 at 9:16 pm
to anon1234, the military rounds are generally slower but they do make for interesting calculations anyway. most “sniper” ammunition is modified from match ammunition. military snipers take the advice of competition shooters. if you are interested in rounds the one that excites me the most is a non-traditional sniper caliber. mk262 mod 1 is a 77 grain .223 cal. bullet with a bc of .372 which is amazing. at the highest altitude in afghanistan it will not go transonic until about 950 meters. also has great terminal ballistics though it was not designed with that in mind.
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