Question

When a monochromatic light source moves toward an observer, its wavelength appears to be shorter than the value measured when the source is at rest. Does this contradict the hypothesis that the speed of light is the same for all observers? Explain.

Short answer

The meter stick is travelling at 0.952c relative to the observer

Step-by-step solution

Concept of the length contraction

The formula for length contraction is$\mathbf{l}\mathbf{=}{\mathbf{l}}_{\mathbf{0}}\sqrt{\mathbf{1}\mathbf{-}\frac{{{\mathbf{u}}_{\mathbf{2}}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}$

Calculate the speed

Here, the proper length${\mathrm{l}}_0$is 1.00 m, and the length of the stick at the observer's frame I is 0.3048m By applying the formula for length contraction we have ,

$$\begin{gathered} l=l_0\sqrt{1-\frac{{u_2}^2/}{c^2}} \\ u=c\sqrt{1-\frac{l^2}{l_0^2}} \\ =c\sqrt{1-\frac{0.{3048}^2}{1.{00}^2}} \\ =0.952c \end{gathered}$$

Thus, the meter stick is travelling at 0.952c relative to the observer