A particle of rest mass M moving with speed 'Beta-not' collides inelastically with (i.e. 'sticks to') a stationary particle of rest mass m. Find the speed of the composite particle. (Answer will contain Beta-not, gamma-not, M and m).
Find the rest energy of the composite particle of the previous particle in terms of the kinetic energy T of the initial particle and the masses M and m.
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Calculations is special relativity can in many cases be simplified by working with quantities that transform covariantly under Lorentz transformations. In this case, we can try to avoid tedious calculations by working with four-momenta. If we work in units such that c = 1, then a particle with energy E and momentum P has a four momentum of (E, P). The P in here has three components (it is just the ordinary momentum vector), and E is the energy, so the four-momentum has four components, hence it's name.
For two four vectors (a0,a1,a2,a3) and (b0,b1,b2,b3) the inner product is defined differently than in case of ordinary vectors. It is defined as:
(a0,a1,a2,a3) dot (b0,b1,b2,b3) = a0 b0 - [a1b1 + a2 b2 + a3b3] (1)
The inner product defined in this way is invariant under Lorentz transformations. Note that some books define the inner product that differs from this definition by an overall minus ...
A detailed solution is given.