 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2015/s7/2/2/solution

p = γm \overset{r}{˙}

, where

γ = (1 - \overset{r}{˙}^{2} / c^{2})^{- 1/2}

. Eliminating velocity gives

H = γm c^{2} = m^{2} c^{4} + c^{2} p^{2} .

\overset{r}{˙} = c^{2} p / H

and

\overset{p}{˙} = 0

. In the limit

∣ p ∣ ≪ m c

H = m c^{2} + p^{2} / (2 m) + O (p^{4} / (m^{3} c^{2}))

, and

\overset{r}{˙} ≃ p / m

. The additive rest energy does not change the classical free-particle equations.

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