Question

A rod lies parallel to the $x$ axis of the reference frame $S$, moving along this axis at a speed of $0.630c$. Its rest length is 1.70 m. What will be its measured length in frame $S$ ?

Short answer

The measured length in frame $S$ is $1.32m$.

Step-by-step solution

The relativistic formula

According to the relativistic formula, the length of the object measured another frame is $l=l_0\sqrt{1-\frac{v^2}{c^2}}$ . Here, $l_0$ is the length of the object lies parallel to the $x$ axis of reference of other frame and $v$ is the velocity of the object.

Use the formula 

Here, the velocity of the rod along the axis of reference frame $S$ is $v=0.630c$. The length of the rod at rest is $l_0=70$.

So, the measured length in the frame $S$ can be calculated as follows:

$$\begin{gathered} l=l_0\sqrt{1-\frac{v^2}{c^2}} \\ l=1.70\sqrt{1-\frac{{\left(0.630c\right)}^2}{c^2}} \\ l=1.70\sqrt{1-{\left(0.63\right)}^2} \\ l=1.32 \end{gathered}$$

Thus, the measured length in the frame $S$ is$1.32m$ .