Question

The rest distance between Patna and Delhi is 1000 km. A nonstop train travels at 360 km h⁻¹. (a) What is the distance between Patna and Delhi in the train frame?
(b) How much time elapses in the train frame between Patna and Delhi?

Answer 1

Given:
Rest distance between Patna and Delhi, L₀ = 1000 km = 10⁶ m
Speed of train, v = 360 km/h$=\frac{360\times 5}{18}=100\mathrm{m}/\mathrm{s}$

(a) Apparent distance between Patna and Delhi is given by

$$\begin{gathered} L\text{'}=L_0\sqrt{1-v^2/c^2} \\ L\text{'}={10}^6\sqrt{1-{\left(\frac{100}{3\times {10}^8}\right)}^2} \\ L\text{'}={10}^6\sqrt{1-\frac{{10}^4}{9\times {10}^{16}}} \\ L\text{'}={10}^6\sqrt{1-\frac{1}{9}\times {10}^{-12}} \\ L\text{'}=56\times {10}^{-9}\mathrm{m} \end{gathered}$$

Change in length = 56 nm

So, the distance between Patna and Delhi in the train frame is 56 nm less than 1000 km.

(b) Actual time taken by the train is given by

t = $\frac{L_0}{v}=\frac{1000\times {10}^3}{100}={10}^4\mathrm{s}=\frac{500}{3}\min$

Change in time, $∆t=\frac{∆L}{v}=\frac{L\text{'}}{v}=\frac{56\times {10}^{-9}}{100}=0.56\times {10}^{-9}\mathrm{s}=0.56\mathrm{ns}$

So, 0.56 nm less than $\frac{500}{3}$ min elapse in the train frame between Patna and Delhi.