Question

Anna and Bob are both born just as Anna's spaceship passes Earth at $0.9c$. According to Bob on Earth planet Z is a fixed away. As Anna passes planet Z on her continuing onward journey, what will be

(a) Bob's age according to Bob

(b) Bob's age according to Anna

(c) Anna's age according to Anna

(d) Anna's age according to Bob

Short answer

(a) When Anna reached planet Z, Bob's age according to Bob is $\text{33.33\hspace{0.17em}y}$.

(b) When Anna reached planet Z, Bob's age according to Anna is$\text{6.33\hspace{0.17em}y}$.

(c) When Anna reached planet Z, Anna's age according to Anna is$\text{14.53\hspace{0.17em}y}$.

(d) When Anna reached planet Z, Anna's age according to Bob is $\text{14.53\hspace{0.17em}y}$.

Step-by-step solution

Given data:

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

The distance of planet Z from Earth with respect to Bob is $d=30\text{\hspace{0.17em}ly}=30c\text{\hspace{0.17em}y}$.

Time dilation and length contraction:

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)

(a) Determining Bob's age according to Bob:

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

$\begin{array}{c}t=\frac{d}{v}\\ =\frac{30c\text{\hspace{0.17em}y}}{0.9c}\\ =33.33\text{\hspace{0.17em}y}\end{array}$

Thus, Bob's age according to Bob is $33.33\text{\hspace{0.17em}y}$.

(b) Determining Bob's age according to Anna:

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

$\begin{array}{c}{d}^{\text{'}}=d\sqrt{1-\frac{v^2}{c^2}}\\ =30\text{\hspace{0.17em}ly}\times \sqrt{1-{0.9}^2}\\ =13.08\text{\hspace{0.17em}ly}\end{array}$

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

$\begin{array}{c}{t}^{\text{'}}=\frac{d^{\text{'}}}{v}\\ =\frac{13.08c\text{\hspace{0.17em}y}}{0.9c}\\ =14.53\text{\hspace{0.17em}y}\end{array}$

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

$\begin{array}{c}t=t^{\text{'}}\sqrt{1-\frac{v^2}{c^2}}\\ =14.53\text{\hspace{0.17em}y}\times \sqrt{1-{0.9}^2}\\ =6.33\text{\hspace{0.17em}y}\end{array}$

Hence, Bob's age according to Anna is $6.33\text{\hspace{0.17em}y}$.

(c) Determining Anna’s age according to Anna:

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

$\begin{array}{c}{t}^{\text{'}}=\frac{d^{\text{'}}}{v}\\ =\frac{13.08c\text{\hspace{0.17em}y}}{0.9c}\\ =14.53\text{\hspace{0.17em}years}\end{array}$

Hence, Anna’s age according to Anna is $14.53\text{\hspace{0.17em}years}$.

(d) Determine Anna's age according to Bob:

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 $14.53\text{\hspace{0.17em}years}$.