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MassEnergy Equivalence and
Relativistic Inelastic Collisions
Jason Harlow and David M. Harrison
Department of Physics
University of Toronto
Introduction
Einstein was led to massenergy 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 simpleminded 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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 Fall '10
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