The equation that is applicable to this problem is expressed as
p = γmu
where p is the relativistic momentum, m is equal to the proton mass and u is the speed of proton. γ is equal to 1/√(1 - u²/c²) where c is the speed of light.
Another equation to determine p fully is to use this relativistic equation:
E = γmc²
where E is the total kinetic energy including the gamma energy.
We are given with E = 2.7 × 10^(-10) J and from a proton’s data, mc^2 = 938.27201 MeV. Using the conversion from MeV to J which is 1 eV = 1.60218 * 10^(-19) J, we compute for γ
γ=E/mc^2 =2.7 × 10^(-10) J / 938.27201x10^6 eV *(1.60218 * 10^(-19) J/eV) = 1.80
we use this number to determine u from the second equation γ = 1/√(1 - u²/c²)
we square both sides and cross multiply, resulting to
γ² (1 - u²/c²) = 1
u²/c² = (γ² -1)/γ²
u = c/y sq rt of (γ² -1)
u = 0.83
Substituting eventually to the first equation, we get
p = γmu
p = 1.80 * 1.67262 * 10^(-27) kg * 0.83 * 2.99792 * 10^8 m/s
p = 7.481 * 10^(-19) kg.m/s
