Question

In Fig. 37-9, observer S detects two flashes of light. A big flash occurs at ${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{1200}\mathbf{}\mathbf{m}$and , slightly later, a small flash occurs at ${\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{480}\mathbf{}\mathbf{m}$. The time interval between the flashes is$\mathbf{∆}\mathbf{t}\mathbf{=}{\mathbf{t}}_{\mathbf{2}}\mathbf{-}{\mathbf{t}}_{\mathbf{1}}$. What is the smallest value of $\mathbf{∆}\mathbf{t}$ for which observer S^' will determine that the two flashes occur at the same x^' coordinate?

Short answer

The required value of $∆\mathrm{t}$ is $2.4\mathrm{\mu s}$.

Step-by-step solution

Describe the expression of time delay between two events

The time delay between two events is given by,

$\mathbf{∆}\mathbf{t}\mathbf{=}\frac{\mathbf{v}\mathbf{∆}{\mathbf{x}}^{\mathbf{\text{'}}}}{{\mathbf{c}}^{\mathbf{2}}\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{v}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}}$

As$\left|v\right|\mathbf{\to }\mathbf{c}$,

The time delay between two events is given by,

$\mathbf{∆}\mathbf{t}\mathbf{=}\frac{\mathbf{∆}{\mathbf{x}}^{\mathbf{\text{'}}}}{\mathbf{c}}$ ……. (1)

Determine the value of  

Find the value of $∆{\mathrm{x}}^{\text{'}}$.

$\begin{array}{rcl}∆{\mathrm{x}}^{\text{'}}& =& 1200\mathrm{m}-480\mathrm{m}\\ & =& 720\mathrm{m}\end{array}$

Substitute 720 m for $∆{\mathrm{x}}^{\text{'}}$, and $3\times {10}^8\mathrm{m}/\mathrm{s}$for c in equation (1).

$\begin{array}{rcl}∆\mathrm{t}& =& \frac{720\mathrm{m}}{3\times {10}^8\mathrm{m}/\mathrm{s}}\\ & =& 2.4\times {10}^{-6}\mathrm{s}\\ & =& 2.4\mathrm{\mu s}\end{array}$

Therefore, the required value of $∆\mathrm{t}$is $2.4\mathrm{\mu s}$.