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Physics of sniping

May 21st, 2008 · 55 Comments

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.

Tags: Physical Concepts

55 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!

    Matt replies: Sounds like a good suggestion! I’ll do that in a future post. That’s a cool bit of information I didn’t know about ENIAC. Another similar thing I’ll post about eventually are the mechanical calculators used to calculate trajectories for aiming battleship guns in WWII. They were quite sophisticated since at those ranges they also had to take into account the curvature of the earth and Coriolis effects.

  • 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.

    Matt replies: That’s what I meant. A trajectory is certainly at least implicit in a firing solution in any case. ;)

    That thing about the optimum launch angle is really interesting! I think I may put the battleship post on hold and work on that one instead. Probably model the v2 air resistance term as containing a coefficient proportional to the density of the atmosphere… ~e-ch or similar, where h is the altitude and c is an appropriate constant. (Actually I see there’s a standard approximation which should do the trick). At that point probably a numeric Mathematica solution will be the easiest way to show good results. I’ll try to have it up this week if I have time. Thanks for the suggestion!

  • 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


  • 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.

  • 12 asdfasdf // Jun 23, 2009 at 6:32 pm


  • 13 Name // Dec 21, 2009 at 11:40 pm

    About the 1.5 mile shot. Everthing that could possibly increase the distance of the projectile was done to the bullet. The bullets were left in the sun to increase the burn temperature during the expansion of the powder. The shot was from an elevation of 9000 ft in the mountains. The sniper actually ran out of his canadian made ammo and was using some american made ammo. The sniper was successful on his third shot at the person holding the RPK machine gun. The first shot was adjusted by his spotter from the vapor trail left by the bullet. The splash was low and right. The second shot hit the man’s backpack. A third shot and a direct hit. Longest confirmed kill.

  • 14 Sgt K // Jan 20, 2010 at 9:04 pm

    The Acronym D.O.P.E. stands for Data On Personal Equipment. It’s used for maintenance and us used to log your current situation in the field. It contains range cards shots and range usage times. No sniper want’s to get caught with a book containing all of his previous locations and shots.

  • 15 Sgt K // Jan 20, 2010 at 9:11 pm

    BTW .223 isn’t used by the govit’s 5.56 X 45 there is a difference. also .223 is designated marksman sniper systems such as the M24(Remington 700) M26(M14) and newer M110(AR-10) all shoot .308 which is 7.62X51 NATO. It has much longer range and better takedown power. The 5.56 Designated marksman is for intermediate range at best around 3-500 meters at most. The NATO 7.62 X 51 can reach out and touch someone at ranges over 800M. The rifle the Canadian was using shot the 50 B.M.G. guaranteed.

  • 16 Ross Mowery // Feb 24, 2010 at 10:49 am

    The Canadian sniper was using a .50 bmg caliber rifle; specifically, a Mcmillan tac-50.

  • 17 ssgt Savage // May 14, 2010 at 3:34 pm

    With the experience i’ve had shooting, the mathematical equations and visual concepts are useful, however, in the end it comes down more to you and your rifle

  • 18 Deity // Jun 22, 2010 at 10:31 am

    Just an update chaps, new record was set by a british sniper with an AI AWSM in .338 Lapua Magnum. (L115A3)

  • 19 Ghost Sniper // Sep 13, 2010 at 1:01 pm

    The earth,s magnetic field changes each day/night and again changes with sun spots.. the rest of the many numbers involved in ultra long shots can be calculated, however slowing the ticker down in extreem stress situations is not an art that one can learn from a book, at that very moment it is a reflection of the madness required in the mind of the sniper who knows the real risks involved, yet has the ability to switch off when “required” to do so.

    Ghost Sniper.

  • 20 sam campbell // Sep 28, 2010 at 1:46 am

    1.5 mile shot is not the longest confirmed shot I own ten miles of land an i hunt on it 3 miles are cleared off an i have a shootin rang set up. with my 30.06 winchester an i have recorded this dozens of times i shoot nickels quarters half dollars an 1 time have i ever accidently hit a dime that was because my arm slipped an squeezed the trigger. at an approximate 1 3/4 of a mile i started shootin scorin direct hits steadily.
    its been a year since ive shot them shots cause im now hittin nickels at 2 5/8 of a milean to prove i a have video footage of every shot i have made which include at least 10-20 hits per video.

  • 21 mnash // Oct 4, 2010 at 5:40 pm

    Firstly, Sam Campbell I would love to see your video and tear it apart. If you can’t even write with correct grammar I doubt you can come close to comprehending the necessary equations to make such shots. Now, ;back to what I wanted to comment on. I saw an interesting show on the Military Channel the other day that brought up a new method of sniping that I thought would be an interesting topic for Matt (the author of the article) to write on. It involved shooting through a loop-hole that is no larger than a few millimeters than the bullet itself. This method, of course, would provide the sniper greater cover against enemies, such as counter-snipers. The show wouldn’t release the formula for performing shots like these but I’d thought it would be an intriguing subject to look in to. Also, the Cheytac M200 “Intervention” and its revolutionary .408 round would make another interesting story.

  • 22 Jay M. // Apr 4, 2011 at 11:44 am


    The Canadian Sniper Rob Furlong, whose shot in Afghanistan was taken in March 2002; took the shot with a .50 caliber round not .308. It was with a McMillan Tac-50 Sniper Rifle, however, due to availability of .50 caliber rounds in Afghanistan the Canadians were actually using the U.S. made .50 caliber rounds (logistics to Afghanistan can be tricky sometimes with Joint Ops). These American rounds are a bit “hotter” then the standard Canadian .50 cal load. It actually took Furlong 3 shots to neutralize his target (1 missed, 1 hit targets backpack and of course 1 kill)

    Also of note: Master Cpl. Perry’s shot which was taken only a week prior with U.S. made .50 cal round (also an official “confirmed kill”) which was only 120m short of Furlongs record shot.

    As Deity pointed out, Furlong’s record was broken and still stand at 2707 yds. This shot was taken by British Sniper CoH Craig Harrison with a L115A3 Rifle chambered in .338 Lapua Magnum.

    There has been A TON of misinformation about these shots/records online. Just wanted to make sure as mathematicians we ensured the quality of our data ;). Even more so as a veteran and former shooter; I wanted to make sure we honor those men by keeping the stories straight. I am sure they would appreciate that.

  • 23 igvaka2445 // Aug 28, 2011 at 8:06 pm

    You are right those men did hit the target got the kill and we should never try to take that away from them they mad the kill thats what counts.

  • 24 Eric // Aug 31, 2011 at 8:37 am

    hey i’m about to enter my senior year of high school. i know i want to major in physics to help with shooting for when i enter the special forces. but i am not sure what area of physics i need to study. can someone help me>?

  • 25 diablo iii release date // Sep 1, 2011 at 7:14 am

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  • 26 shonak // Dec 9, 2011 at 11:29 am

    can anybody tell me how mathematics is involved in sniping?

  • 27 mason watkins // Mar 13, 2012 at 9:32 am

    it is involved by the time the bullet leaves the barral to the time of impact. it shows how many feet the shot came.

  • 28 Night crawler // Aug 7, 2012 at 1:47 am

    Thanks for correcting Sam because, WOW. Clearly has never made a long distance shot in his life. I haven’t heard of the British snipers shot. However if he got over 1.5 down range with a 308 lapua, that was an amazing shot. Much more impressive to me then the Canadian with the tac 50.
    Next I would like to clarify some things. First of all, at ranges over 1.25 miles, everything comes into effect. Ambient temp. Bore and receiver temp. Temp of the rounds. Elevation above sea level. Barometric pressure. And yes the Coriolis effect. Coriolis effect doesn’t come into play with very short events, such as water draining from a sink or tub. The event itself is not large enough or long enough to affect the out comes. However, with very long distance shots, the laws of exterior ballistics are paramount. Why? Because the event takes place at very high speeds and over a long enough period of time. In the case of the Canadian sniper, everything played right into his hand. The only reason he took the shot was the perfect conditions involved. He was elevated. Minimal wind. Perfect temps all around. Shooting from one hill, over a small valley, to another hill. The fact that the tango’s had nothing with enough range to shoot back etc… And yes he was shooting to the west And yes with very long shots, shooting to the west gives you slightly more distance. Why? Yep, the Coriolis effect. The earth spins at about 1000 mph.. After a four second flight time that translates to approximately 1/4 mile extra range. It would be four times that if the earth did not “drag it’s atmosphere with it.”. But in this perfect storm, it gave the Canadian the extra length he needed.
    Last but not least, bullets do not start dropping the second they come out of the barrel. Anybody who has been to U.S. Army basic training will tell you, that the round comes out of the barrel and rises slightly. In an m4 or m16 round (5.56 NATO) it rises and crosses the sight line at approximately 200-250 meters. So when shooting a target at 100 meters a slodier has to aim lower then the spot they wish to hit or the round will be high. As it has not crossed the sight line yet. But after 200-250 meters we have to adjust our aim point higher as the round has crossed the sightline and will be low. Also the tac50 system with the “hotter” American rounds at 50 calibre twist due to rifleing. And as their flight becomes more and more unstable due to the effects of time, distance and medium traveled through, and the other factors mentioned above, tend to go up and to the right, as that is the direction of the twist. All of this adds up to an unbelievable, once in a generation shot. Kuddo’s to the young man who made that shot. And to the spotter who helped dope that scope. And if a British shooter beat that shot with a 308, well then my hat is off to him.

  • 29 Modern Furniture // Sep 20, 2012 at 12:50 pm

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  • 30 mtch // Nov 26, 2012 at 8:59 pm

    To night crawler agree with your perspective just thought I’d help to clarify that the British shooter used a.338 lapua magnum not the .308 described in your comment

  • 31 Mojo // Jan 6, 2013 at 10:02 pm

    NightCrawler has a brilliant grasp on the science and real world reality. However referring the British shot, that record was surpassed in early 2012, when an unknown Australian sniper recorded a kill at 2815 meters (3079 yards) using a .50 BMG M82A1[5].
    The shot is now part of manufacturer simulations and SASR training.

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  • 35 William R. DkilleSiley // Apr 3, 2013 at 1:00 pm

    Just looked on Youtube after reading this article and accompanying posts.
    There is a Sam Campbell on there, but looks like his kill shots are in video game format.

  • 36 William R. DeSilvey // Apr 3, 2013 at 1:01 pm

    ………..hate it when the cursor skips on this laptop.
    A good physics solution would involve a 12 ga slug.

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  • 38 One Inch Arty // May 18, 2013 at 3:18 pm

    A 1.5+ mile shot in the larger scheme of war is like pissing on a forest fire. You know it’s not going to help much, but you do it anyway, just for the hell of it.

  • 39 Wendy Galarza // Aug 19, 2013 at 8:30 pm

    I would have being very happy with a printed formulas to calculate the air resistance, also with a initial lunch altitude, initial velocity , and so on..
    but it was cute anyway.

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  • 42 Joe // Oct 25, 2013 at 3:01 am

    @One Inch Arty: Can’t say I agree. The right shot on the right target can be more effective than an entire artillery barrage. But it’s all about the intel.

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