What is the magnitude of the relativistic momentum of a proton with a relativistic total energy of 2.7 3 10210 j?

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