We calculate the range of a projectile experiencing air resistance in the asymptotic region of large velocities by introducing the Lambert $W$ function. From the exact solution for the range in terms of the Lambert $W$ function, we derive an approximation for the maximum range in the limit of large velocities. Analysis of the result confirms an independent numerical result observed in an introductory physics class that the angle at which the maximum range occurs, $\theta max,$ goes rapidly to zero for increasing initial firing speeds $v0\u226b1.$ We show that $\theta max\u223c(ln\u200av0)/v0.$

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Educational aids
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If we apply the bisection method to find $W$ such that $f(W)=W\u200aexp(W)\u2212z=0,$ we should distinguish two regions of real $z.$ For $z>0,$ there is always a root in $0<W<z,$ so the initial interval that brackets the root can always be safely set to $[0,z].$ (For large $z,$ the upper limit ought be set to $ln\u200az$ to avoid overflow.) For $\u2212e\u22121<z<0$ (region of interest here), there is one root in $\u22121<W<0$ so the initial bracket can be set to [−1,0]. (Another root in $[ln(\u2212z),\u22121]$ is of no interest here.) For $z<\u2212e\u22121,$ there are an infinite number of complex roots, one that diverges to $ln(\u2212z),$ while the others converge to zero. The complex roots have no physical significance to the problem discussed here.

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© 2004 American Association of Physics Teachers.

2004

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