Problem 19.47 Challenge

จาก \(v_{g}=\frac{dE}{dp}\)

\(v_{g}=\frac{dp^{2}c^{2} + E_{0}^{2}}{dp}\)

\(v_{g}=\frac{2pc^{2}}{2p^{2}c^{2}+E_{0}^{2}}\)

\(v_{g}=\sqrt{\frac{p^{2}c^{4}}{p^{2}c^{2} + E_{0}^{2}}}\)

\(v_{g}=\sqrt{\frac{\gamma ^{2}m_{0}^{2}v^{2}c^{4}}{\gamma ^{2}m_{0}^{2}v^{2}c^{2} + m_{0}^{2}c^{4}}}\)

\(v_{g}=\sqrt{\frac{c^{2}}{1 +\frac{c^{2}}{\gamma ^{2}v^{2}}}}\)

\(v_{g}=\sqrt{\frac{c^{2}}{1 +\frac{c^{2} - v^{2}}{v^{2}}}}\)

\(v_{g}=\sqrt{\frac{c^{2}}{\frac{v^{2} + c^{2} - v^{2}}{v^{2}}}}\)

\(v_{g}= v \;\blacksquare\)