Question

An electron of $\mathit{\beta }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9999987}$moves along the axis of an evacuated tube that has a length of 3.00 m as measured by a laboratory observer S at rest relative to the tube. An observer S’ who is at rest relative to the electron, however, would see this tube moving with speed v ( $\mathbf{=}\mathit{\beta }\mathit{C}$).What length would observer S’ measure for the tube?

Short answer

The length measured by observer S’ for the tube is $0.0153m$.

Step-by-step solution

Identification of given data

The speed parameter is $\beta =0.999987$$\beta =0.999987$

The length of the evacuated tube is $L_0=3m$

The length contraction is used to find the length measured by the observer S’ for the tube.

Determination of length measured by observer S’ for the tube

The length measured by observer S’ for the tube is given as:

$L=L_0\sqrt{1-{\beta }^2}$

Substitute all the values in the equation.

$$\begin{gathered} L=\left(3m\right)\sqrt{1-{\left(0.999987\right)}^2} \\ L=0.0153m \end{gathered}$$

Therefore, the length measured by observer S’ for the tube is $0.0153m$.