Verifying Equation for Particle Energy (E): p=γpmv

[seto6]

so we have total energy of particle ->E

p=γ_p=1/(1-(v/c)²)^-1/2

1)E=γ_pmc²=E₀+K=Rest energy+ kinetic energy
=mc²+mv²/2
the second line correct?

2) E²-(pc)²=E₀
so P= mv or p=γ_pmv

im sure that we are suppose to use p=γ_pmv but not sure some one verify pls

 

[mathman]

I can't quite understand 2).

However for 1), if you expand p in a power series in (v/c)², the first two terms are what you are showing. The higher order terms matter when v/c -> 1.

 

[yuiop]

so we have total energy of particle ->E

p=γ_p=1/(1-(v/c)²)^-1/2

1)E=γ_pmc²=E₀+K=Rest energy+ kinetic energy
=mc²+mv²/2
the second line correct?

Nope, the second line is not correct. mv2 is the Newtonian expression for kinetic energy (KE).
You can obtain the correct relativistic expression for KE from:

γ=(1-(v/c)²)^-1/2

E=γmc²=E₀+KE = Rest energy+ kinetic energy

KE = E-E₀ = γmc²-mc² = mc²(γ-1)

2) E²-(pc)²=E₀
so P= mv or p=γ_pmv

im sure that we are suppose to use p=γ_pmv so but not sure some one verify pls

You should be using p = γmv = mv(1-(v/c)²)^-1/2 so that

E²-(pc)²=E₀²

E²=E₀²+(pc)²

E²=(mc²)²+(γmvc) ²