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You are trapped in a round coliseum, and a gladiator with a spear is chasing you. You cannot defend your self, however you may run quicker than the gladiator.
- You run at 11 ft per second, and the gladiator runs at 10 ft per second.
- The gladiator’s spear has a spread of 10 ft. You should keep at the very least 10 ft away from the gladiator, or else you may be killed.
- The gladiator is just not sensible sufficient to take a strategic path that optimally chases you down. They solely head straight in your path.
- You management the beginning place of your self and the gladiator.
How large does the coliseum should be for you to have the ability to keep away from the gladiator indefinitely?
Generalized, in case your velocity is V, the gladiator’s velocity is W, and the spear size is X, how large does the coliseum should be?
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Normalise models so the gladiator runs at velocity $1$, you run at velocity $s>1$ and the spear has size $d$. The thought is to have the gladiator $G$ run in a circle $Gamma_1$ of some radius $r$ centred within the coliseum’s origin $O$. You, $Y$, put your self on the finish of the vector of size $d$ pointing away from $G$ within the path of his instantaneous velocity (tangent to $Gamma_1$) and run in a bigger circle $Gamma_2$ concentric with $Gamma_1$.
Then $triangle OGY$ is right-angled at $G$ with $OG=r,GY=d$ and therefore $OY=sqrt{r^2+d^2}$. You wish to keep the right-angled triangle’s form and maintain it spinning round $O$; since velocity is proportional to radius, to match you and $G$‘s most speeds you want$$frac{sqrt{r^2+d^2}}r=sqrt{1+(d/r)^2}=s$$ $$r=frac d{sqrt{s^2-1}}$$and a coliseum of dimension $rs$. For the numerical values right here with $s=1.1$ and $d=1$ this provides $r=frac{10}{sqrt{21}}=2.1821789dots$ ($21.821dots$ ft) and the coliseum radius as $frac{11}{sqrt{21}}=2.4003967dots$ ($24.003dots$ ft).
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