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Two people A and B start at a certain point. B runs away from A to another point. A shoots arrows at B.

Is it helpful for A to run in a zigzag rather than in a constant direction?


enter image description here


To be more precise:

Suppose A starts and remains at $(0,-1)$, and B starts at $(0,0)$ and has to get to $(0,d)$ where $d > 0$ or some point $(b,c)$ sufficiently close to it ie $d((b,c),(0,d)) < \epsilon$ for $\epsilon > 0$

Suppose we have two options, and let $a,v > 0$

  1. Running straight from $(0,0)$ to $(0,d)$ at a speed $v$

  2. Running from $(0,0)$ to $(a,a)$ to $(0,2a)$ to $(-a,3a)$ to $(0,4a)$ to ... at a speed $v$

Under what conditions, if any, is option 2 better than option 1? What might be some other important physical factors to consider?


Context:

This is based on a recent episode of an ongoing TV series named

Game of Thrones

A clip can be found in the link below

https://www.youtube.com/watch?v=9n71Rzgs7lE

I have read some people saying that if character B

ie Rickon Stark

ran in a zigzag or at least in some kind of $S$-shaped path

character B would have had a better chance of not being hit by the arrows of character A

ie Ramsay Bolton.


Of course there are some differences in my given scenario and that part of the episode such as not having $a, v, d$ but rather $a_t, v_t, d_t$ which are positive functions.

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    $\begingroup$ I'm voting to close this question as off-topic because (a) it isn't clear what the question is asking about physics and (b) it's blindingly obvious that it's harder to hit a target with two degrees of freedom than a target with just a single degree of freedom. $\endgroup$ – John Rennie Jun 21 '16 at 15:57
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    $\begingroup$ I disagree with @JohnRennie. I think the question is one of kinematics, and is just as "on topic" as typical "pursuit" questions which ask where A catches up with B if A sets off later but with greater velocity. Actually I think it is quite a good open-ended question : the challenge is to select key factors and simplifying assumptions which keep the problem manageable yet realistic. The question is "Under what conditions is option 2 preferable?" I suggest that investigating the opposite (conditions for option 1 to be preferable) makes the problem more narrowly focussed. $\endgroup$ – sammy gerbil Jun 21 '16 at 17:45
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    $\begingroup$ BTW congratulations on a well presented question : succinct title, inclusion of context, and a mathematical model. $\endgroup$ – sammy gerbil Jun 23 '16 at 12:20
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    $\begingroup$ @BCLC : Not at all. Not a perfect question, but better than many. I'm disappointed with the On Hold decision. Seems a popular question. $\endgroup$ – sammy gerbil Jun 23 '16 at 16:33
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    $\begingroup$ I think it's a wonderfully constructed question. Those who have closed it never have asked any question at all! As if they are omniscient. $\endgroup$ – Deschele Schilder Sep 1 at 23:38
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The zig-zag strategy seems trivially obvious - but it might not be the better strategy in a particular situation. I suggest rather than asking under what conditions this strategy is preferable, you ask under what conditions the counter-intuitive straight-line strategy is preferable.

The advantage of zig-zagging is that it presents a smaller "collision cross-section" both in terms of target area (side-on) and time (an arrow following the runner remains "on target" longer). The advantage of the straight-line path is that it gets the runner out of range in the shortest time. It also minimises the relative speed of impact, thus minimising damage - though in realistic scenarios this is unlikely to be significant.

The question needs further specification, in particular regarding the speed $c$ of the arrows, the time $\tau$ between shots (which includes reloading and aiming), and their range $R$.

I presume that B cannot try to dodge the arrows as they fly. But there are many other factors to consider. For example :
* How much of a head start is B given?
* How accurate (in terms of angular spread) is the archer's aim?
* Does B follow a predetermined zig-zag path known to A (as in your conditions), or are his turns made at random?
* Is gravity to be taken into account? The disadvantage of the straight-line path is then reduced as B gets further from A, requiring the arrow to be aimed away from the target and into the sky.
* What cover is available? Reaching cover might be less risky than getting out of range.
* The relative cross-sections presented by B to hits from the back, side and above. This could include any body armour or exposure of weak points.

Complications could be piled on indefinitely to make the mathematical model more realistic. Judicious simplifications are necessary, as in your model, to avoid unnecessary effort towards spurious accuracy. But without the minimum specifications suggested above ($c$, $\tau$, $R$) it is not possible to decide which strategy is best.

Generally: In the unrealistic case that the speed of arrows $c$ is less than the speed of the runner $v$, the direct route is clearly preferable. Even when $c > v$ the direct route might be preferable if it takes B out of range much quicker, presenting significantly fewer opportunities for shots to be taken.

Presumably B is given a "head start". This time is best used running directly away from A, and only then (possibly) zig-zagging. Likewise, while A is reloading B should run directly away from A.

Developing your model : I assume the 'head start' is the time $\tau$ between shots.

  1. For the straight-line option, arrows are fired at times $n\tau$ where $n=1, 2, 3,...$, and reach B at times $t_n = n\tau \frac{c}{c-v}$. B is then distances $y_n=vt_n$ from A. The number N of shots arriving within range is given by $y_N < R$.

I shall assume that the probability of hitting a target is proportional to the solid angle which it presents. Suppose A is certain to hit B if face-on and within a distance $y_0$. Then the probability of hitting B at distance $y$ is approx. $(\frac{y}{y_0})^2$ provided that $y>y_0$. So for this strategy the probability of B being killed is
$P_1 = \Sigma^N_1 (\frac{y_n}{y_0})^2$.

I assume that all hits are ultimately fatal. More realistically you could assign a probability of a hit being fatal; if non-fatal it reduces B's speed by a certain factor.

  1. For the zig-zag option at angle $\theta$ to the straight-line direction, the speed of separation of B from A is now approx. $v_y=v\cos\theta$. To simplify calculations, I assume the length of the zig-zags is much smaller than the distance AB. Arrows reach B at times $t_n = n\tau \frac{c}{c-v_y}$. B is then distances $y'_n=v_y t_n$ from A. The number M of shots arriving within range is given by $y'_M < R$.

Since B is now slightly side-on to the arrows the target area is smaller by a factor of approx. $\cos\theta$. This assumes that B is a 'cardboard cut-out'. In realistic situations this factor could be ignored, except perhaps for shots from above.

The probability of hitting B is now approx. $(\frac{y'_n \cos\theta}{y_0})^2$. Using the zig-zag strategy, the probability of B being killed is
$P_2 = \Sigma^M_1 (\frac{y'_n \cos\theta}{y_0})^2$.

I have assumed that, as for the straight line strategy, B's position $B_1$ when the arrow lands can be predicted by A. If B can change direction at a random time between shots arriving (keeping the same angle $\theta$), $B_1$ will be one of two points reachable from his position $B_0$ when the arrow was released. If angle $\theta$ can be changed also at random (but always $<90$ degrees), $B_1$ will be somewhere on a semi-circle of radius $vT$ centred on $B_0$; $T$ is the time of flight of the arrow. The new value of $\theta$ affects the next shot arriving at $t_{n+1}$. In both cases the uncertainty in position $B_1$ can be included as a further probability factor.

Another refinement would include the effect of gravity on the arrow. This alters time of flight, range and effective target area, which is smaller from above.


Conclusion : In many cases one strategy (probably zig-zagging, especially at random) will be clearly the better one. In some cases which strategy is the better might depend in a very complicated way on the various factors involved. The view that zig-zagging is not always the best strategy is supported in the following article relating to escape from a gunman :

http://www.activeresponsetraining.net/dont-run-in-a-straight-line-and-other-bad-advice

In the Game of Thrones, zig-zagging probably was better, but not clearly so. In the 1st two shots Ramsay seems to be 'toying' with Rickon. In the 3rd he looks more serious. The final shot appears to pierce Rickon's heart. Was that luck or skill? If it was skill, I think Rickon had little chance from the start. Maximising distance from Ramsay (by doing as he did) was probably Rickon's best chance, at least until Ramsay started to take the 'game' seriously. Ramsay would have relished the challenge of hitting a zig-zagging target. Whatever Rickon's strategy, Ramsay would not have allowed Rickon to get so far that he had a significant chance of surviving.

I think the best strategy for Rickon was to mix strategies : first run directly away, then zig-zag as the shots got closer to target.

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  • $\begingroup$ sammy gerbil, what do you think of reopening? we need 3 more please. PS: my thanks must've been deleted for being too chatty $\endgroup$ – BCLC Sep 2 at 7:57

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