Consider a collision between two protons that is perfectly inelastic: an incident proton with mass m(p), kinetic energy K, and momentum magnitude p combines with a target proton that was initially stationary to form a single product particle with mass M.
Now,
In this case, the problem is alleviated by using colliding beams.
A pair of interacting particles' combined momentum may be zero.
After the collision, the center of mass is at rest.
Therefore, all the initial kinetic energy can be used to create particles.
Let K be the kinetic energy of each of the two identical colliding particles.
Therefore, the energy released by the two colliding beams are:
E₁ = K + mc² and E₂ = K + mc²
In the final state,
E(f) = Mc² where M is the new mass of the single product after the collision.
By conservation of energy,
E₁ + E₂ =E(f)
K + mc² + K + mc² = Mc²
2K + 2mc² = Mc²
Taking 2mc² common, we get
Mc² = 2mc²[ 1 + (K/mc²)]
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