91Ó°ÊÓ

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

The length of the rod is $L$.

The speed parameter is $\mathrm{β}$.

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 $\mathit{γ}$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,

$L=\frac{L_0}{\mathrm{γ}}$ …… (i)

The expression for the Lorentz factor is given by,

$\mathrm{γ}=\frac{1}{\sqrt{1-{\mathrm{β}}^2}}$

Substitute $\frac{1}{\sqrt{1-{\mathrm{β}}^2}}$for$\mathrm{γ}$ into equation (i).

$L=L_0\sqrt{1-{\mathrm{β}}^2}$

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

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