91Ó°ÊÓ

Speed of Anna with respect to Bob is$v=0.9c$.

The distance of planet Z from Earth with respect to Bob is $d=30\text{ l²â}=30c\text{ y}$.

The time interval$t^{\text{'}}$ measured by a moving observer at velocity $v$is related to the same time interval$t$ measured by a stationary observer as

$t^{\text{'}}=\frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$ ..... (1)Here,$c$ is the speed of light in a vacuum.

The length$l^{\text{'}}$measured by a moving observer at velocity$v$ is related to the same length$l$ measured by a stationary observer as

$l^{\text{'}}=l\sqrt{1-\frac{v^2}{c^2}}$ .....(2)

The time taken for Anna to reach planet Z according to Bob is

$\begin{array}{c}t=\frac{d}{v}\\ =\frac{30c\text{ y}}{0.9c}\\ =33.33\text{ y}\end{array}$

Thus, Bob's age according to Bob is $33.33\text{ y}$.

From equation (2), distance to planet Z from Earth according to Anna is

$\begin{array}{c}{d}^{\text{'}}=d\sqrt{1−\frac{v^2}{c^2}}\\ =30\text{ l²â}×\sqrt{1−{0.9}^2}\\ =13.08\text{ l²â}\end{array}$

Planet Z is moving towards Anna at$0.9c$according to Anna. Time taken for planet Z to reach Anna according to Anna is

$\begin{array}{c}{t}^{\text{'}}=\frac{d^{\text{'}}}{v}\\ =\frac{13.08c\text{ y}}{0.9c}\\ =14.53\text{ y}\end{array}$

This is Anna's age according to Anna. But according to Anna, Bob is moving away with velocity$0.9c$.So Bob's age according to Anna from formula (1) will be

$\begin{array}{c}t=t^{\text{'}}\sqrt{1−\frac{v^2}{c^2}}\\ =14.53\text{ y}×\sqrt{1−{0.9}^2}\\ =6.33\text{ y}\end{array}$

Hence, Bob's age according to Anna is $6.33\text{ y}$.

Time taken for planet Z to reach Anna according to Anna is

$\begin{array}{c}{t}^{\text{'}}=\frac{d^{\text{'}}}{v}\\ =\frac{13.08c\text{ y}}{0.9c}\\ =14.53\text{ y±ð²¹°ù²õ}\end{array}$

Hence, Anna’s age according to Anna is $14.53\text{ y±ð²¹°ù²õ}$.

As per man on earth, Anna’s clock is slower, so Anna looks younger than the time she passes bob. So, by applying time dilation you get,

$\begin{array}{c}d\text{'}=vt\\ t=\frac{d\text{'}}{v}\\ =\frac{13.0\text{8ly}}{\left(0.9c\right)}\\ =14.53\text{years}\end{array}$

Hence, Anna’s age according to Bob is $14.53\text{ y±ð²¹°ù²õ}$.