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I was surfing on Instagram, and I found this amazing proto

enter image description here

whose description is "the probability of such an event to happen is incredibly small, so this is a really curious finding".

Well.. I'm interested in some formula to compute a possible probability for such an event to happen, considering two shot bullets (even of different kinds).

According to the result, I may assume those two bullets have been shot from two guns whose directions were like orthogonal (but maybe they weren't).

I may assume the probability may depends upon the velocity of the two bullets, namely either upon they masses and upon the kind of guns (but I don't understand how this data can enter in the equation.. maybe we can think about the power?).

Any idea?

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  • $\begingroup$ To compute a "probability", you need to specify some sort of initial conditions together with a probability density over which to average, otherwise this is deterministic classical mechanics which knows no "probability" for something happening. Also, this is extremely open to interpretation, and you can't really expect to compute anything here except ballpark estimates that highly depend on the chosen model. $\endgroup$
    – ACuriousMind
    Feb 26, 2016 at 23:32
  • $\begingroup$ @ACuriousMind Well, you have two guns in orthogonal direction, which shoot at the same time. There will be a single point in space in which the 2 bullets can collide, and this depends upon the considered factors above (velocity, friction, power.. maybe masses too..) so I guess one may start with thinking about those factors? $\endgroup$
    – Les Adieux
    Feb 26, 2016 at 23:35
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    $\begingroup$ Of course that's where you start thinking. But the question is just too vague to be a good fit for the SE model. Like, which variables do we consider as degrees of freedom and which as fixed? What is the sample space upon which you want to compute the probability? Are all values for the parameters uniformly likely? Do the bullets always penetrate when they meet? and so on... $\endgroup$
    – ACuriousMind
    Feb 26, 2016 at 23:40
  • $\begingroup$ For starters you should explore the possibility of this happening, at all. Fire a bullet at a resting bullet and see what happens. I am not sure that colliding bullets will behave this way. It might very well have been faked (e.g. by firing from near distance at a bullet that we held in place by something). $\endgroup$
    – CuriousOne
    Feb 26, 2016 at 23:47
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    $\begingroup$ The photo looks 'shopped'. It seems very unlikely that two colliding bullets would end up entwined as shown in that photo. $\endgroup$
    – Gert
    Feb 26, 2016 at 23:51

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The answer depends on many factors you did not specify, so let's look at the variables that go into this.

  1. The bullets have to travel along an intersecting trajectory
  2. They have to arrive at the same time

I will assume that if these two conditions are met, they will penetrate (although that requires very different hardness from the two bullets; in fact, if you look at the picture I would highly suspect that one bullet was going significantly faster than the other...)

Re. item 1, you have to aim your guns to the same point, within about a fraction of the diameter of the bullet, $\delta$. Depending on the distance $D$ to the point where they meet, this requires an angular precision of $\alpha=\frac{\delta}{D}$. If you know that your gun is held steady with a Gaussian angular distribution with a standard deviation of $\sigma$, then the probability that each gun is pointing in the right direction at the same time is

$$P_{aim}=\frac{1}{\sqrt{2\pi\alpha}}\int_{\alpha/2}^{\alpha/2}e^{-x^2/2\sigma}dx$$

The probability that both are aimed the same way is the square of that. To be more precise, they could both aim "a little high" or "a little low", and still hit each other, so the real calculation is a bit more complicated. Below, I will (for the purpose of estimating) use a uniform probability distribution - the assumptions we are making as sufficiently outrageous that that introduces an insignificant additional error.

Next, we have to fire the bullets at the same time. If the velocity of a bullet is $v$, and the bullet has to be in the right place within a factor $\Delta x$, then the timing of the two shots has to be within $\frac{\Delta x}{v}$.

Whether you can get the timing that close depends on how hard you try, obviously. If you can assume that with a bit of practice, two people can fire a gun within 1/20th of a second of each other (think about musicians - they can stay in time better than that), then the probability of firing within that time interval is approximately $\frac{\delta t \Delta x}{v}$. In this case, you don't need to take the square - because the first bullet can be fired at any time, you just have to fire the second one "within a short interval".

Now there are some real people out there who can shoot very, very well. See for example Bob Munden. Not only can he draw and fire a gun in 1/50th of a second, he can hit a silver dollar in mid-air - which gives us an upper bound on that "angular uncertainty" we were talking about.

If we use his shooting skills as an example, then for a bullet with a length of 15 mm traveling at a velocity of 400 m/s you need a positional accuracy of about 4 mm (to be near the middle of the 15) and a timing precision of 10 µs. That means that if his draw gets him within 0.02 s, he would need about 2000 shots to hit that target - for timing. If we further assume his aim is "good enough to hit a silver dollar every time", and we need precision to within 3 mm, then his aim would be good enough roughly 1 in 10 times. Now the other gun would have to also be right, but if we assume a uniform distribution of the two angular distributions (rather than the Gaussian distribution I mentioned above) then you have 10 opportunities for each of the two trajectories to intersect - so still a 1 in 10 probability over all.

Combining these, the probability (if two very skilled gunmen tried to do this) would be around 1 in 20,000.

But this is just a sample calculation. If you set this up with guns on tripods, carefully controlled charges in your cartridge, and electronic triggers, you might be able to get much, much closer. Something for mythbusters to try...

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