91Ó°ÊÓ

The time difference between the two events,

$\begin{array}{l}\mathrm{Δ³Ù}={\mathrm{t}}_{\mathrm{B}}−{\mathrm{t}}_{\mathrm{A}}\\ \mathrm{Δ³Ù}=1\text{‰ÓÔ¼}\end{array}$

The distance,

$\begin{array}{l}\mathrm{Δ³\ae }={\mathrm{x}}_{\mathrm{B}}−{\mathrm{x}}_{\mathrm{A}}\\ \mathrm{Δ³\ae }=240\text{″¾}\end{array}$

The expression to calculate the Lorentz factor is given as follows.

$\mathbf{γ}\mathbf{=}\sqrt{\frac{\mathbf{1}}{\mathbf{1}\mathbf{−}{\mathbf{β}}^{\mathbf{2}}}}$ …(¾±)

The equation to calculate the expression for the${\mathbf{Δ³\ae }}^{\mathbf{\text{'}}}$ is given as follows.

${\mathbf{Δ³\ae }}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{γ}\mathbf{(}\mathbf{Δ³\ae }\mathbf{−}\mathbf{βcΔ³Ù}\mathbf{)}$ …(¾±¾±)

Here, $\mathbf{γ}$is the Lorentz factor and$\mathbf{c}$ is the speed of light.

(a)

Calculate the expression for${\mathrm{Δ³\ae }}^{\text{'}}$.

Calculate the expression for ${\mathrm{Δ³Ù}}^{\text{'}}$.

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

$\begin{array}{l}{\mathrm{Δ³\ae }}^{\text{'}}=\frac{1}{\sqrt{1−{\mathrm{β}}^2}}(\mathrm{Δ³\ae }−\mathrm{βcΔ³Ù})\\ {\mathrm{Δ³\ae }}^{\text{'}}=\frac{\mathrm{Δ³\ae }−\mathrm{βcΔ³Ù}}{\sqrt{1−{\mathrm{β}}^2}}\end{array}$

Substitute$240\text{″¾}$ for$\mathrm{Δ³\ae }$ ,$3×{10}^8\text{″¾}/\text{s}$ for$\mathrm{c}$ and$1\text{‰ÓÔ¼}$ for$\mathrm{Δ³Ù}$ into equation (i).

$\begin{array}{l}{\mathrm{Δ³\ae }}^{\text{'}}=\frac{240\text{″¾}−\mathrm{β}×3×{10}^8\text{″¾}/\text{s}×1×{10}^{−6}\text{ s}}{\sqrt{1−{\mathrm{β}}^2}}\\ {\mathrm{Δ³\ae }}^{\text{'}}=\frac{\left(240\text{″¾}\right)−\left(300\text{″¾}\right)\mathrm{β}}{\sqrt{1−{\mathrm{β}}^2}}\end{array}$ …(¾±¾±¾±)

Hence the expression for the${\mathrm{Δ³\ae }}^{\text{'}}$ is $\frac{\left(240\text{″¾}\right)−\left(300\text{″¾}\right)\mathrm{β}}{\sqrt{1−{\mathrm{β}}^2}}$.

(b)

The graph between${\mathrm{Δ³\ae }}^{\text{'}}$ and$\mathrm{β}$ is shown below.

(c)

The graph between${\mathrm{Δ³\ae }}^{\text{'}}$ and $\mathrm{β}$is shown below.

Calculate the value of $\mathrm{β}$.

Substitute$0$ for${\mathrm{Δ³\ae }}^{\text{'}}$ into equation (iii).

$\begin{array}{c}\frac{240−300\mathrm{β}}{\sqrt{1−{\mathrm{β}}^2}}=0\\ 240−300\mathrm{β}=0\\ 300\mathrm{β}=240\\ \mathrm{β}=0.8\end{array}$

Hence the value of $\mathrm{β}$is$0.8$ .