One-dimensional collisions and Chebyshev polynomials
Abstract
This paper investigates a sequence of inelastic collisions resulting among three spheres, whose centers are on a line, when at the outset the leftmost sphere is propelled to the right and the other two spheres are at rest. Explicit expressions are obtained for velocity, time, and distance variables, and a simple method is given for determining the total number of collisions which take place. Under certain circumstances this number is infinite; in this case the final state is shown to be achieved in finite time and to be essentially the same as for the completely inelastic case. The Chebyshev polynomials of the second kind are prominent in the analysis. The results are applied to generalize work of R. H. Romer concerning the motion of air-carts on a track.
- Publication:
-
American Journal of Physics
- Pub Date:
- March 1977
- DOI:
- 10.1119/1.11003
- Bibcode:
- 1977AmJPh..45..255T
- Keywords:
-
- 03.20.+i;
- 02.30.Gp;
- Special functions