Question

The diagram shows Bob's view of the passing of two identical spaceship. Anna's and his own, where ${\mathbf{\gamma }}_{\mathbf{v}}\mathbf{=}\mathbf{2}$. The length of either spaceship in its rest frame is . What are the readings on Anna', two unlabelled clocks?

Short answer

The unlabelled readings from Anna’s clocks are $\left(-\frac{2{\mathrm{vL}}_{\mathrm{o}}}{{\mathrm{c}}^2}\right)$and $\left(-\frac{6{\mathrm{vL}}_{\mathrm{o}}}{{\mathrm{c}}^2}\right)$.

Step-by-step solution

Identification of the given information

The given data can be written as:

• The Lorentz factor for problem is ${\mathrm{\gamma }}_{\mathrm{v}}=2$.
• The length of spaceship is data-custom-editor="chemistry" ${\mathrm{L}}_{\mathrm{o}}$.

Significance of Lorentz factor

The Lorentz factor is essential for calculation related to the relativistic motion of particles. The readings for Anna’s calculation are calculated by linear transformations of time dilation.

Determination of the first reading in Anna’s clock

The first reading in Anna’s clock can be given as:

${\mathrm{t}}_1={\mathrm{\gamma }}_{\mathrm{v}}\left(-\frac{\mathrm{vx}}{{\mathrm{c}}^2}+\mathrm{t}\right)$

Substitute all the values in the above equation.

$$\begin{gathered} {\mathrm{t}}_1=\left(2\right)\left(-\frac{\mathrm{v}\left({\mathrm{L}}_{\mathrm{o}}\right)}{{\mathrm{c}}^2}+0\right) \\ {\mathrm{t}}_1=-\frac{2{\mathrm{vL}}_{\mathrm{o}}}{{\mathrm{c}}^2} \end{gathered}$$

Determination of the second reading in Anna’s clock

The second reading in Anna’s clock can be given as:

${\mathrm{t}}_2={\mathrm{\gamma }}_{\mathrm{v}}\left(-\frac{\mathrm{vx}}{{\mathrm{c}}^2}+{\mathrm{t}}_1\right)$

Substitute all the values in the above equation.

$$\begin{gathered} {\mathrm{t}}_2=\left(2\right)\left(-\frac{\mathrm{v}\left({\mathrm{L}}_{\mathrm{o}}\right)}{{\mathrm{c}}^2}+\left(-\frac{2{\mathrm{vL}}_{\mathrm{o}}}{{\mathrm{c}}^2}\right)\right) \\ {\mathrm{t}}_2=-\frac{6{\mathrm{vL}}_{\mathrm{o}}}{{\mathrm{c}}^2} \end{gathered}$$

Therefore, the unlabelled readings from Anna’s clocks are $\left(-\frac{2{\mathrm{vL}}_{\mathrm{o}}}{\mathrm{c}}\right)$and $\left(-\frac{6{\mathrm{vL}}_{\mathrm{o}}}{\mathrm{c}}\right)$.