Measured at rest, a rod has an exact length of 1 m . Determine its length when it travels at a speed of 3 10^5 m s ⁻¹ .

Step-by-step solution

The rest length of the rod is L₀ = 1 m . The speed of the rod is v = 3 10^5 m s ⁻¹ . The speed of light is c = 3 10^8 m s ⁻¹ . Using the length contraction formula: L = L₀ 1 - v^2 c^2 Substituting the given values: L = 1 1 - ( 3 10^5 3 10^8 )^2 L = 1 - (10⁻³)^2 L = (1 - 10⁻⁶)^ 1/2 Using the binomial approximation (1 - x)^n 1 - nx for x 1 : L 1 - 1 2 10⁻⁶ L 1 - 0.0000005 L = 0.9999995 m