91影视

The given data is:

• The distance of the planet Y is $\mathrm{x}=5\mathrm{ly}$.
• The speed of spaceship is $\mathrm{v}=0.8\mathrm{c}$.
• The duration for occurrence of event is $\mathrm{t}=2\mathrm{yr}$.

The above problem is based on the concept of Lorentz transformations and time dilation. All the distances in light-year should be changed to meters for calculation.

The formula to calculate the distance of planet Y is given as:

$\mathrm{d}=\frac{\mathrm{x}-\mathrm{vt}}{\sqrt{1-{\displaystyle \frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}}}}$

Substitute all the values in the above equation,

$$\begin{gathered} \mathrm{d}=\frac{\left(5\mathrm{ly}\right)\left({\displaystyle \frac{9.461脳{10}^{15}\mathrm{m}}{1\mathrm{ly}}}\right)-\left(0.8\mathrm{c}\right)\left({\displaystyle \frac{3脳{10}^8\mathrm{m}/\mathrm{s}}{\mathrm{c}}}\right)\left(2\mathrm{yr}\right)\left({\displaystyle \frac{3.154脳{10}^7\mathrm{s}}{1\mathrm{yr}}}\right)}{\sqrt{1-{\displaystyle \frac{{\left(0.8\mathrm{c}\right)}^2}{{\mathrm{c}}^2}}}} \\ \mathrm{d}=53.61脳{10}^{15}\mathrm{m} \\ \mathrm{d}=\left(53.61脳{10}^{15}\mathrm{m}\right)\left(\frac{1\mathrm{ly}}{9.461脳{10}^{15}\mathrm{m}}\right) \\ \mathrm{d}=5.67\mathrm{ly} \end{gathered}$$

Therefore, the distance of planet Y according to Anna is data-custom-editor="chemistry" $5.67\mathrm{ly}$.

The time of explosion of planet Y can be given as:

$\mathrm{T}=\frac{\left(-{\displaystyle \frac{\mathrm{vx}}{{\mathrm{c}}^2}}+\mathrm{t}\right)}{\sqrt{1-{\displaystyle \frac{{\mathrm{v}}^2}{{\mathrm{c}}^2}}}}$

Substitute all the values in the above equation,

$$\begin{gathered} \mathrm{T}=\frac{-{\displaystyle \frac{\left(0.8\mathrm{c}\right)\left({\displaystyle \frac{3脳{10}^8\mathrm{m}/\mathrm{s}}{\mathrm{c}}}\right)\left(5\mathrm{ly}\right)\left({\displaystyle \frac{9.461脳{10}^{15}\mathrm{m}}{1\mathrm{ly}}}\right)}{{\left(\left(0.8\mathrm{c}\right)\left({\displaystyle \frac{3脳{10}^8\mathrm{m}/\mathrm{s}}{\mathrm{c}}}\right)\right)}^2}}+\left(2\mathrm{yr}\right)\left({\displaystyle \frac{3.154脳{10}^7\mathrm{s}}{1\mathrm{yr}}}\right)}{\sqrt{1-{\displaystyle \frac{{\left(0.8\mathrm{c}\right)}^2}{{\mathrm{c}}^2}}}} \\ \mathrm{T}=-22.33脳{10}^7\mathrm{s} \\ \mathrm{T}=\left(-22.33脳{10}^7\mathrm{s}\right)\left(\frac{1\mathrm{yr}}{3.154脳{10}^7\mathrm{s}}\right) \\ \mathrm{T}=-7.07\mathrm{yr} \end{gathered}$$

Here, the negative sign indicates the explosion on planet Y, which occurs before explosion observed by Anna, so time is negative.

Therefore, the time of explosion is data-custom-editor="chemistry" $7.07\mathrm{yr}$.