New Orleans, Louisiana
June 26, 2016
June 26, 2016
August 28, 2016
Engineering Physics & Physics
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.
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
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