Tensors and Relativity: Chapter 2

Example 2.1

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Next: Example 2.2 Up: Relativistic dynamics Previous: Relativistic dynamics

As a simple example, consider two particles with rest masses tex2html_wrap_inline1436 and tex2html_wrap_inline1438 , moving with speeds tex2html_wrap_inline1440 and tex2html_wrap_inline1442 tex2html_wrap_inline1444 , which collide and coalesce [ see Figure 2.2 ]. What is the mass M and speed U of the resulting particle.

figure517
Figure 2.2: The colliding particles in Example 2.1.

The initial four- momenta are

equation1081

where tex2html_wrap_inline1450 and

equation1083

The final four- momentum is

equation1085

Conservation of four- momentum gives

equation1089

and squaring this relation gives

equation1093

Now

equation1095

and

equation1097

This gives

equation1099

For tex2html_wrap_inline1452 and tex2html_wrap_inline1454 , tex2html_wrap_inline1456 which is the usual Newtonian result. From the x and t components of the four- momentum equation we have

eqnarray1101

Dividing we get

equation1103

For tex2html_wrap_inline1452 and tex2html_wrap_inline1454 we obtain

equation1105