Dimensional analysis, falling bodies, and the fine art of not solving differential equations

American Journal of Physics

Volumn 72, Issue 4, 2004, Pages 534-537

  Bohren, Craig F ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1842764869     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1574042     Document Type: Article

References (12)

1. F. W. Sears, M. W. Zemansky, and H. D. Young, University Physics (Addison-Wesley, Reading, MA, 1982), 6th ed.
2. R. A. Serway, Physics for Scientists and Engineers with Modern Physics (Saunders, Philadelphia, 1992), 3rd ed.
3. D. Halliday and R. Resnick, Fundamentals of Physics (Wiley, New York, 1988), 3rd ed.
4. D. C. Giancoli, Physics (Prentice-Hall, Englewood Cliffs, NJ, 1995), 4th ed.
5. J. R. Strutt (Lord Rayleigh), "On the light from the sky, its polarization and colour," Philos. Mag. 41, 107-120 (1871). This paper and other seminal papers on atmospheric light scattering are reprinted in Scattering in the Atmosphere, edited by C. F. Bohren (SPIE Optical Engineering Press, Bellingham, WA, 1989).
6. J. R. Strutt (Lord Rayleigh), "On the light from the sky, its polarization and colour," Philos. Mag. 41, 107-120 (1871). This paper and other seminal papers on atmospheric light scattering are reprinted in Scattering in the Atmosphere, edited by C. F. Bohren (SPIE Optical Engineering Press, Bellingham, WA, 1989).
7. For a delightful exposition of the outlook of a "real" mathematician see G. H. Hardy, A Mathematician's Apology (Cambridge U.P., Cambridge, 1969), pp. 121-124.
8. 2, so the linear drag law is invalid for such objects during all but a tiny fraction of their trajectories. This law is valid, however, for cloud droplets, which have diameters of order 10 μm. A rule of thumb is that if you can readily see a falling body, the linear drag law is not applicable to it.
9. For the consequences of a variable density with height to the motion of a body dropped at rest in Earth's atmosphere see P. Mohazzabi and J. H. Shea, "High altitude free fall," Am. J. Phys. 64, 1242-1246 (1996).
10. H. Schlichting, Boundary-Layer Theory (McGraw-Hill, New York, 1968), 6th ed., p. 17.
11. The solution here for v and z is essentially the same as that given by G. Feinberg, "Fall of bodies near the Earth," Am. J. Phys. 33, 501-502 (1965).
12. C. G. Adler and B. L. Coulton, "Galileo and the Tower of Pisa experiment," Am. J. Phys. 46, 199-201 (1978).