Dot Physics

There were a couple of things that bothered me about the MythBusters’ myth where they fired bullets in the air. The myth was that a bullet fired in the air could kill you. The first problem is that it is not a myth. There are several reported cases of people being killed from bullets that were fired in the air. The Mythbusters tested this by finding out how fast a bullet would be going if fired straight up. A couple of problems:

  • First, they measured the terminal velocity of a tumbling bullet, not a spinning one. I really don’t know how long a bullet will stay spinning, but I guess this could be a problem.
  • Second, they really only looked at bullets that were fired straight up, not at an angle. I think this makes a big difference.

So, what am I going to do about it? First, let me say again I really liked this particular MythBusters episode. They offered up some real experimental data – in particular when they fired the 9 mm rounds straight up and measured the time of flight. Yes, there were some problems, but that is what makes the MythBusters so cool.

In my last look at the bullets fired up, I made a numerical calculation for a bullet with air resistance. A couple of points about my model (which may indeed be flawed, but it’s my best shot).

  • I used an air resistance that was proportional to the square of the velocity.
  • Using data from wikipedia, I have a coefficient of drag that changes with the bullet speed – based on radar data.
  • Did not model a change in the gravitational field (it just didn’t change that much for the heights I was looking at)
  • I DID look at change in the density of air at different heights

With that model, my calculations were fairly similar to the data from MythBusters. So, to proceed, I ran the calculations for bullets shot at different angles. Fired straight up agrees with my initial program (which I did just to make sure I didn’t mess anything up when I changed it to 2-D). Another thing I did to test for reasonableness of the model was to look at near horizontal firing. Please note that I really don’t know much about guns. The first time I shot a gun (ever) was just last year. So, my near horizontal is shot at 1 degree above horizontal from a height of 1.5 meters. Here is a plot of the trajectory:

i-df53f2a08cee0b56346620d6698093e0-3006_1degree.jpg

This is the .30-06 with an initial velocity of 880 m/s. I don’t know if this is accurate, but I don’t see anything wrong with it. Good enough to proceed. Now I am going to run the calculation starting at zero degrees and increasing 1 degree up to 90. I will plot the speed the bullet has when it hits the ground. For the .30-06, here is what I get:

i-2a7d3878e2c0bb2ac6abacab7b54d660-306_finalspeedplot.jpg

So, why does it do that? If you shoot the bullet straight horizontal, then it is not going to be in the air as long and the air drag will not have as much time to do negative work on the bullet and slow it down. Note that the air resistance force is always in the opposite direction to the displacement so that the work (done by air on the bullet) will be negative. When the bullet is shot straight up, it should have time to reach terminal velocity on the way back down. Also, it is useful to think about the case where there is no air resistance, in this case all the bullets would have the same final speed regardless of angle.

I don’t know at what speed a bullet is lethal, but you can see that the final speed drops off rapidly with angle. If I had to guess, I would say that a firing angle of 45 degrees could be considered by some as still “shooting in the air”. Also, the lower the angle the larger horizontal distance the bullet would travel. Think about when you see people firing off ‘celebration’ shots. Sometimes these low angles do happen.

What about for the 9mm round? Here is the same calculation with the 9 mm.

i-13912dd285f9081fd69b1b1a4dfff1b9-9mm_finalspeed_vs_angle.jpg

One last note. I tried to use my model to determine the range of a fired bullet. However, I am not sure how range is defined? Is it from a standing position? Is the bullet fired completely horizontal? Who knows.

Oh, and here is the program if it makes you happy – bulletimpact2d.py

Comments

  1. #1 Glen Thomas
    October 5, 2009

    Interesting model and the curves look realistic. A couple of points..

    The spin stabilization of the bullet is likely to be maintained only for a few (3 – 5?) seconds, as any more will take the bullet beyond the accurate or lethal range, and would reduce the muzzle velocity unnecessarily by gripping the barrel too much.

    (To answer your question, range will be defined according to the intended role of the weapon – a target shooter is just looking at predictability of the trajectory, while a soldier will need penetration at range. In either case, the transition from super- to sub-sonic will mess up the aerodynamics of the bullet, so the range will be the supersonic range for military rifles.)

    The bullet would also not have the ‘weather-cock’ stability of a feathered arrow. The gyroscopic effects would keep the bullet pointing in the same, original, direction even as the path curved. As the bullet spins down it will start to precess (as the Army’s 5.56 mm bullets are supposed to do on contact with a target) and rapidly move to tumbling as the center of pressure moves forwards with subsonic speeds.

    A bullet fired upwards will not come down pointing downwards (or, indeed, pointing upwards still), so the Mythbuster team got this bit right .

  2. #2 Rhett
    October 5, 2009

    @Glen,

    Thanks for the useful info. I guess the ballistics radar data I used doesn’t go down to low speeds where the bullet would tumble. I guess I could use the MythBuster’s data to estimate this low-speed, tumblinging coefficient of drag and add it to the table. I just may do that.

  3. #3 Ali
    October 10, 2009

    “The first problem is that it is not a myth. There are several reported cases of people being killed from bullets that were fired in the air.”

    Haha. Isn’t this the standard preface when propagating myth/folklore? Citations please! 🙂

  4. #4 Bill
    December 1, 2009

    Here is a story about several people being shot on New Years Day in Dallas. It happens every year.
    http://www.dallasnews.com/sharedcontent/dws/news/localnews/stories/DN-newyearincidents_02met.ART.State.Edition2.17b0d.html

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