MassEnergyEquivalence

# MassEnergyEquivalence - Mass-Energy Equivalence and...

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Mass-Energy Equivalence and Relativistic Inelastic Collisions Jason Harlow and David M. Harrison Department of Physics University of Toronto Introduction Einstein was led to mass-energy equivalence by considering the interaction between a charged particle and an electromagnetic field. 1 His original argument is fairly complex for beginning students. A few years ago one of us (DMH) devised a simpler way of demonstrating that the mass of a body must increase with its speed. The description considers a totally inelastic collision between two equal masses and only requires knowing about conservation of momentum and that relativistic speeds do not add in a simple way. It has proved to be effective with Physics students; it also works well in liberal arts courses that use little or no mathematics. However, when we tried to make this description quantitative we were surprised to discover that a simple-minded approach did not work. It turns out that to conserve momentum in all frames it must be assumed that the kinetic energy lost during the collision is converted to mass. We will first describe the qualitative argument. We then go through the quantitative analysis and will associate the dissipated energy with the mass of the system. The Qualitative Argument Figure 1 shows two objects with equal rest masses undergoing a perfectly inelastic collision as viewed from a reference frame in which the total momentum is zero. We will call this Frame 1. Before the collision the speed of each object is 0.6 c, and after the collision the two objects are stationary. All the initial kinetic energy has been converted to energy of heat and deformation. Figure 2 shows the same collision in a frame of reference moving to the left at 0.6 c relative to the first frame. We call this Frame 2. In this frame, before the collision object 1 is moving at a speed given by the relativistic formula for addition of velocities: 2 / 1 2 ' c vu v v + = (1)

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## This note was uploaded on 10/30/2010 for the course MP 200 taught by Professor Guna during the Fall '10 term at Duke.

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MassEnergyEquivalence - Mass-Energy Equivalence and...

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