How much work is required to accelerate a proton from rest up to a speed of 0.987 c?

For the work-energy theorem, the work done to accelerate the proton is equal to the variation of kinetic energy of the proton:

where  is the initial kinetic energy, which is zero because initially the proton is at rest.

We need to find the kinetic energy of the proton: we can't use the classical formula  since the proton is at relativistic speed. Therefore, we must use the relativistic formula:

where 
m is the proton mass
v is its final speed
c is the speed of light

Substituting , v=0.987 c and , we find


and this is equal to the work needed to accelerate the proton up to a speed of 0.987c.

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