Problem 16.35 Challenge

จาก \(\Delta t_{0} = 1,00 hr = 3,600 s\)

และ \(v = 1,000 km/hr = 278 m/s\)

จาก \(Δt = γΔt_{0}\)

\(\Delta t =\frac{\Delta t_{0}}{1 -\frac{v^{2}}{c^{2}}}\)

\(_{-1/2}\)

\(\Delta t = \Delta t_{0}(1 -\frac{v^{2}}{c^{2}})\)

\(_{n}\)

จาก\((1\pm x)\approx 1\pm nx\) เมื่อ\(x\ll 1\) และ\(x=\frac{v^{2}}{c^{2}}\)

จะได้ \(Δt \approx Δt_{0}(1 + \frac{v^{2}}{2c^{2}})\)

\(\Delta t \approx \Delta t_{0} +\frac{v^{2}\Delta t_{0}}{2c^{2}}\)

\(\therefore \Delta t - \Delta t_{0} \approx \frac{v^{2}\Delta t_{0}}{2c^{2}}\)

\(\Delta t - \Delta t_{0} \approx \frac{(278 m/s)^{2}(3,600 s)}{2(3.00\times 10^{8} m/s)^{2}}\)

\(\Delta t - \Delta t_{0} \approx 1.54 ns \;\blacksquare\)