Revisiting the One-Dimensional Elastic Collision of Rigid Bodies on a Frictionless Surface Using Singularity Functions

Aziz S Inan; Peter M Osterberg

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Conference: 2016 ASEE Annual Conference & Exposition
Location: New Orleans, Louisiana
Publication Date: June 26, 2016
Start Date: June 26, 2016
End Date: June 29, 2016
ISBN: 978-0-692-68565-5
ISSN: 2153-5965
Conference Session: Engineering Physics Technical Session 4
Tagged Division: Engineering Physics & Physics
Page Count: 6
DOI: 10.18260/p.26106
Permanent URL: https://strategy.asee.org/26106
Download Count: 754

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Paper Authors

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Aziz S Inan University of Portland

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Dr. Aziz Inan is a professor in Electrical Engineering at the University of Portland (Portland, OR), where he has also served as Department Chairman. He received his BSEE degree from San Jose State University in 1979 and MS and Ph.D. degrees in electrical engineering from Stanford University in 1980 and 1983 respectively. His research interests are electromagnetic wave propagation in conducting and inhomogeneous media. He is a member of Tau Beta Pi and senior member of IEEE.

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Peter M Osterberg University of Portland

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Dr. Peter Osterberg is an associate professor in Electrical Engineering at the University of Portland (Portland, OR). He received his BSEE and MSEE degrees from MIT in 1980. He received his Ph.D. degree in electrical engineering from MIT in 1995 in the field of MEMS. He worked in industry at Texas Instruments, GTE, and Digital Equipment Corporation in the field of microelectronics. His research interests are microelectronics, MEMS, and nanoelectronics.

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Abstract

Physics and engineering students are taught to formulate the one-dimensional elastic collision problem involving rigid bodies using the well-established energy and momentum formulas. However, the authors note that it is equally instructive to reformulate this problem by full usage of singularity functions involving impulse and step functions. It is the authors’ intent to present this interesting approach which they believe is not fully addressed in the literature.

To demonstrate this approach, the authors consider the well-known one-dimensional elastic collision problem between two rigid objects having different masses approaching each other with different constant velocities on a frictionless surface. The goal is to determine the velocity of each object after collision. At the time of the collision, each object experiences an impulsive force which lasts for an infinitesimal amount of time but with sufficiently large magnitude to allow quick transfer of finite amount of linear momentum between the two bodies. This momentum exchange causes an abrupt “step” in the velocity of each object. The impulsive force acting on each object is mathematically represented in terms of the well-known Dirac delta function (or the impulse function). The abrupt change in the velocity of each object is expressed in terms of the Heaviside unit-step function. By applying the conservation of momentum and the conservation of energy principles, and by incorporating a special singularity integral, the authors derive the well-known expressions for the final velocities of each object.

The authors find this approach very educational and interesting particularly in terms of familiarizing the students with singularity functions and their applications. The authors believe that this formulation would be very useful in formulating different types of elastic collision problems including multi-dimensional collisions.

In addition, the authors emphasize the multi-disciplinary aspect of this approach by providing another brief example involving the calculation of energy stored in a capacitor due to a step-type voltage applied across it.

Citation
Format

Inan, A. S., & Osterberg, P. M. (2016, June), Revisiting the One-Dimensional Elastic Collision of Rigid Bodies on a Frictionless Surface Using Singularity Functions Paper presented at 2016 ASEE Annual Conference & Exposition, New Orleans, Louisiana. 10.18260/p.26106

TY - CPAPER
AB - Physics and engineering students are taught to formulate the one-dimensional elastic collision problem involving rigid bodies using the well-established energy and momentum formulas. However, the authors note that it is equally instructive to reformulate this problem by full usage of singularity functions involving impulse and step functions. It is the authors’ intent to present this interesting approach which they believe is not fully addressed in the literature.

To demonstrate this approach, the authors consider the well-known one-dimensional elastic collision problem between two rigid objects having different masses approaching each other with different constant velocities on a frictionless surface. The goal is to determine the velocity of each object after collision. At the time of the collision, each object experiences an impulsive force which lasts for an infinitesimal amount of time but with sufficiently large magnitude to allow quick transfer of finite amount of linear momentum between the two bodies. This momentum exchange causes an abrupt “step” in the velocity of each object. The impulsive force acting on each object is mathematically represented in terms of the well-known Dirac delta function (or the impulse function). The abrupt change in the velocity of each object is expressed in terms of the Heaviside unit-step function. By applying the conservation of momentum and the conservation of energy principles, and by incorporating a special singularity integral, the authors derive the well-known expressions for the final velocities of each object.

The authors find this approach very educational and interesting particularly in terms of familiarizing the students with singularity functions and their applications.
The authors believe that this formulation would be very useful in formulating different types of elastic collision problems including multi-dimensional collisions.

In addition, the authors emphasize the multi-disciplinary aspect of this approach by providing another brief example involving the calculation of energy stored in a capacitor due to a step-type voltage applied across it.

AU - Aziz S Inan
AU - Peter M Osterberg
CY - New Orleans, Louisiana
DA - 2016/06/26
PB - ASEE Conferences
TI - Revisiting the One-Dimensional Elastic Collision of Rigid Bodies on a Frictionless Surface Using Singularity Functions
UR - https://strategy.asee.org/26106
DO - 10.18260/p.26106
ER -