Analytic Approximations of Projectile Motion with Quadratic Air Resistance

Affiliation(s)

^{1}
Department of Administrative Sciences, Metropolitan College, Boston University, Boston, USA.

^{2}
Department of Physics, University of Massachusetts Dartmouth, North Dartmouth, USA.

^{3}
Institute for Theoretical Physics, Vienna University of Technology, Vienna, Austria.

ABSTRACT

We study projectile motion with air resistance quadratic in speed. We consider three regimes of approximation: low-angle trajectory where the horizontal velocity, u, is assumed to be much larger than the vertical velocity w; high-angle trajectory where ; and split-angle trajectory where . Closed form solutions for the range in the first regime are obtained in terms of the Lambert W function. The approximation is simple and accurate for low angle ballistics problems when compared to measured data. In addition, we find a surprising behavior that the range in this approximation is symmetric about , although the trajectories are asymmetric. We also give simple and practical formulas for accurate evaluations of the Lambert W function.

We study projectile motion with air resistance quadratic in speed. We consider three regimes of approximation: low-angle trajectory where the horizontal velocity, u, is assumed to be much larger than the vertical velocity w; high-angle trajectory where ; and split-angle trajectory where . Closed form solutions for the range in the first regime are obtained in terms of the Lambert W function. The approximation is simple and accurate for low angle ballistics problems when compared to measured data. In addition, we find a surprising behavior that the range in this approximation is symmetric about , although the trajectories are asymmetric. We also give simple and practical formulas for accurate evaluations of the Lambert W function.

Cite this paper

nullR. Warburton, J. Wang and J. Burgdörfer, "Analytic Approximations of Projectile Motion with Quadratic Air Resistance,"*Journal of Service Science and Management*, Vol. 3 No. 1, 2010, pp. 98-105. doi: 10.4236/jssm.2010.31012.

nullR. Warburton, J. Wang and J. Burgdörfer, "Analytic Approximations of Projectile Motion with Quadratic Air Resistance,"

References

[1] R. D. H. Warburton and J. Wang, “Analysis of asymptotic projectile motion with air resistance using the Lambert W function,” American Journal of Physics, Vol. 72, pp. 1404–1407, 2004.

[2] S. R. Valluri, D. J. Jeffrey, and R. M. Corless, “Some applications of the Lambert W functions to physics,” Canadian Journal of Physics, Vol. 78, pp. 823–830, 2000.

[3] S. R. Cranmer, “New views of the solar wind with the Lambert W function,” American Journal of Physics, Vol. 72, pp. 1397–1403, 2004.

[4] D. Razansky, P. D. Einziger, and D. R. Adam, “Optimal dispersion relations for enhanced electromagnetic power deposition in dissipative slabs,” Physical Review Letters, Vol. 93, 083902, 2004.

[5] S. M. Stewart, Letters to the Editor, American Journal of Physics, Vol. 73, pp. 199, 2005. “A little introductory and intermediate physics with the Lambert W function,” Proceedings of 16th Australian Institute of Physics, Vol. 005, pp. 194–197, 2005.

[6] E. Lutz, “Analytical results for a Fokker-Planck equation in the small noise limit,” American Journal of Physics, Vol. 73, pp. 968–972, 2005.

[7] P. H?vel and E. Sch?ll, “Control of unstable steady states by time-delayed feedback methods,” Physical Review E, Vol. 72, 046203, 2005.

[8] There is a solution presented in terms of the slope angle by A. Tan, C. H. Frick, and O. Castillo, “The fly ball trajectory: An older approach revisited,” American Journal of Physics, Vol. 55, pp. 37–40, 1987. But the slope angle is unknown except at the top (zero) of the trajectory, and can be found only numerically or graphically. Therefore, the solution is not in closed form.

[9] G. W. Parker, “Projectile motion with air resistance quadratic in the speed,” American Journal of Physics, Vol. 45, pp. 606–610, 1977.

[10] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, Vol. 5, pp. 329–359, 1996.

[11] H. R. Kemp, “Trajectories of projectile motion in air for small times of flight,” American Journal of Physics, Vol. 55, pp. 1099–1102, 1987.

[12] B. Hayes, “Why W?” American Science, Vol. 93, pp. 104–108, 2005.

[13] J. Wang, To be published.

[14] M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions,” Dover, New York, pp. 18, 1970.

[1] R. D. H. Warburton and J. Wang, “Analysis of asymptotic projectile motion with air resistance using the Lambert W function,” American Journal of Physics, Vol. 72, pp. 1404–1407, 2004.

[2] S. R. Valluri, D. J. Jeffrey, and R. M. Corless, “Some applications of the Lambert W functions to physics,” Canadian Journal of Physics, Vol. 78, pp. 823–830, 2000.

[3] S. R. Cranmer, “New views of the solar wind with the Lambert W function,” American Journal of Physics, Vol. 72, pp. 1397–1403, 2004.

[4] D. Razansky, P. D. Einziger, and D. R. Adam, “Optimal dispersion relations for enhanced electromagnetic power deposition in dissipative slabs,” Physical Review Letters, Vol. 93, 083902, 2004.

[5] S. M. Stewart, Letters to the Editor, American Journal of Physics, Vol. 73, pp. 199, 2005. “A little introductory and intermediate physics with the Lambert W function,” Proceedings of 16th Australian Institute of Physics, Vol. 005, pp. 194–197, 2005.

[6] E. Lutz, “Analytical results for a Fokker-Planck equation in the small noise limit,” American Journal of Physics, Vol. 73, pp. 968–972, 2005.

[7] P. H?vel and E. Sch?ll, “Control of unstable steady states by time-delayed feedback methods,” Physical Review E, Vol. 72, 046203, 2005.

[8] There is a solution presented in terms of the slope angle by A. Tan, C. H. Frick, and O. Castillo, “The fly ball trajectory: An older approach revisited,” American Journal of Physics, Vol. 55, pp. 37–40, 1987. But the slope angle is unknown except at the top (zero) of the trajectory, and can be found only numerically or graphically. Therefore, the solution is not in closed form.

[9] G. W. Parker, “Projectile motion with air resistance quadratic in the speed,” American Journal of Physics, Vol. 45, pp. 606–610, 1977.

[10] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, Vol. 5, pp. 329–359, 1996.

[11] H. R. Kemp, “Trajectories of projectile motion in air for small times of flight,” American Journal of Physics, Vol. 55, pp. 1099–1102, 1987.

[12] B. Hayes, “Why W?” American Science, Vol. 93, pp. 104–108, 2005.

[13] J. Wang, To be published.

[14] M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions,” Dover, New York, pp. 18, 1970.