A New Look at Energy — Question 1

Question 1

Energy and momentum jointly determine rest mass and kinetic energy

A particle has total energy $E=5.00\,\mathrm{MeV}$ and momentum such that $pc=4.00\,\mathrm{MeV}$. Use $E^2=p^2c^2+m^2c^4$, $E=\gamma mc^2$, $K=E-mc^2$, and $\mathbf v=c^2\mathbf p/E$.

1. Find its rest energy and kinetic energy.

2. Find its speed and Lorentz factor, checking the mass-shell identity.

Question 1 - Solution

Energy and momentum jointly determine rest mass and kinetic energy

(a) Method and calculation

$$mc^2=\sqrt{E^2-p^2c^2}=\sqrt{25-16}\,\mathrm{MeV}=\boxed{3.00\,\mathrm{MeV}}.$$$$\boxed{K=5.00-3.00=2.00\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{v/c=pc/E=0.800},\qquad\boxed{\gamma=E/(mc^2)=5/3}.$$$$\boxed{E^2-(pc)^2=9.00\,\mathrm{MeV}^2=(mc^2)^2}.$$

Physical check

Total energy includes rest energy. Subtracting rest energy before using the mass-shell equation would give an inconsistent result.