Measured at rest, a rod has an exact length of 1 m . Determine its length when it travels at a speed of 3 10^6 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 10^6 m s ⁻¹ . The speed of light is c = 3 10^8 m s ⁻¹ . According to the special theory of relativity, the contracted length L is given by: L = L₀ 1 - v^2 c^2 Substituting the given values: L = 1 1 - ( 3 10^6 3 10^8 )^2 L = 1 - (10⁻²)^2 L = 1 - 10⁻⁴ Using the binomial approximation (1 - x)^ 1/2 1 - x 2 for x 1 : L 1 - 10⁻⁴ 2 L 1 - 0.00005 L = 0.99995 m