A rod is to move at constant speed$\mathit{v}$along the$\mathit{x}$axis of reference frame $\mathit{S}$, with the rod’s length parallel to that axis. An observer in frame $\mathit{S}$is to measure the length$\mathit{L}$of the rod. Which of the curves in Fig. 37-15 best gives length $\mathit{L}$(vertical axis of the graph) versus speed parameter$\mathit{\beta }$?

The constant speed of the rod is $v$along the $x$axis.

The length of the rod is $L$.

The speed parameter is $\beta$.

The expression to calculate the length of the rod in respect to observer in the frame$S\text{'}$ is given as follows.

$\mathit{L}\mathbf{=}\frac{{\mathbf{L}}_{\mathbf{0}}}{\mathbf{Y}}$

Here, ${\mathit{L}}_{\mathbf{0}}$is the proper length of the rod and $\mathit{\gamma }$is the Lorentz factor.

Let assume a frame $S\text{'}$which is moving with the velocity of $v$and for observer in this frame the length of the rod seems to be shorter which is given by,

$L=\frac{L_0}{\gamma }$ …… (i)

The expression for the Lorentz factor is given by,

$\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$

Substitute $\frac{1}{\sqrt{1-{\beta }^2}}$for$\gamma$ into equation (i).

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

As the value of the$\beta$ increases, $1-{\beta }^2$, therefore value of length decreases.

Hence the curve $c$gives the best graph between length and speed parameter.