Reference frame$\mathit{S}\mathbf{\text{'}}$is to pass reference frame $\mathit{S}$at speed $\mathit{v}$along the common direction of the $\mathit{x}\mathbf{\text{'}}$and $\mathit{x}$axes, as in Fig. 37-9. An observer who rides along with frame $\mathit{s}\mathbf{\text{'}}$is to count off $\mathbf{25}\mathit{s}$on his wristwatch. The corresponding time interval $\mathbf{∆}\mathit{t}$is to be measured by an observer in frame $\mathit{s}$. Which of the curves in Fig. 37-15 best gives $\mathbf{∆}\mathit{t}$(vertical axis of the graph) versus speed parameter$\mathit{\beta }$?

The speed of the reference frame$S\text{'}$ is $v$.

The observer is sitting in the frame $S\text{'}$and count off $25s$on wristwatch.

The expression to calculate the time interval is given as follows.

$\mathbf{∆}\mathit{t}\mathbf{=}\frac{\mathbf{∆}{\mathbf{t}}_{\mathbf{0}}}{\sqrt{\mathbf{1}\mathbf{-}\mathbf{\beta }}}$ …… (i)

Here, $\mathit{\beta }$is the speed parameter.

The expression to calculate the speed parameter is given as follows,

$\mathit{\beta }\mathbf{=}\frac{\mathbf{v}}{\mathbf{c}}$

Here,$\mathit{v}$ is the relative speed of the frame $\mathit{S}\mathbf{\text{'}}$with respect to frame $\mathit{S}$and$\mathit{c}$ is the speed of the light.

Calculate the time interval,

Substitute$25s$for $∆t_0$into equation (i).

$∆t=\frac{25}{\sqrt{1-{\beta }^2}}$

Substitute $\raisebox{1ex}{$v$}\!\left/ \!\raisebox{-1ex}{$c$}\right.$for$\beta$into above equation.

$∆t=\frac{25}{\sqrt{1-{\left(\frac{v}{c}\right)}^2}}$

From the above, as the relative speed of frame$S\text{'}$ is approaches to speed of the light, the value of speed parameter$\beta$ increases, and the value of$1-{\beta }^2$ decreases which result to increases in value of time interval$∆t$ .

Hence the curve $b$is the best representation of the graph between $∆t$and $\beta$.