Question

It is required to cause protons to collide with an energy measured in their centre of mass frame, of 4 M₀c² in excess of their rest energy 2 M₀c². This can be achieved by firing protons at one another with two accelerators each of which imparts a kinetic energy of 2M₀c². Alternatively, protons can be fired from an accelerator at protons at rest. How much energy must this single machine be capable of imparting to a proton? What is the significance of this result for experiments in high energy nuclear physics?

Answer 1

Total energy available in the CMS is 

E^∗ = (m₁² + m₂² + 2E₁m₂)^1/2 

where E₁ is the total energy of projectile of mass m₁ and m₂ is the mass of the target. 

E^∗ = 4M₀ + 2M₀ = 6M₀ 

m₁ = m₂ = M₀ 

6M0 = (M₀² + M₀² + 2E₁M₀)^1/2 

whence E₁ = 17M₀ 

Or T₁ = E₁ − M₀ = 16M₀ 

The significance of this result is that in the colliders a lot more energy is available than in fixed target experiments.