According to Bob, on Earth, it is $\mathbf{20}\mathbf{ly}$ to Planet $\mathbf{Y}$. Anna has just passed Earth, moving at a constant speed $\upsilon$ in a spaceship, When Anna passes Planet $\mathbf{Y}$. She is $\mathbf{20}$years older than when she passed Earth. Calculate $\nu$.

Consider a distance Planet $\text{Y}$ to Earth is${\iota }_0=20\text{\hspace{0.33em}ly}$.

Consider the time taken is $t=20\text{\hspace{0.17em}y}$.

Write the formula of speed of spaceship.

$\nu =\frac{\iota }{t}$ …… (1)

Here, $\iota$ is distance travelled by the spaceship and $\mathbf{t}$ is time taken.

The connection between distance and time can be used to characterise velocity:

$\nu =\frac{d}{t}$ …… (2)

The following equation may be used to show how a relativistic effect causes an object's length to contract:

$\iota ={\iota }_0\sqrt{1-\frac{{\nu }^2}{c^2}}$ …… (3)

A lightyear may be stated as $cy$, which is the speed of light multiplied by the years, as a lightyear is equal to the speed of light in a year. Using Eq. (2), we can describe how far the spaceship has travelled as a result of relativistic effects:

$\begin{array}{c}\iota ={\iota }_0\sqrt{1-\frac{{\nu }^2}{c^2}}\\ =\left(20\text{ly}\right)\sqrt{1-\frac{{\nu }^2}{c^2}}\\ =\left(20\text{cy}\right)\sqrt{1-\frac{{\nu }^2}{c^2}}\end{array}$

It is possible to link the distance and speed using Eqs. (2) and (3). We choose and establish the following phrase to represent the spaceship's speed:

Determine the speed of spaceship.

Substitute $\left(20\text{cy}\right)\sqrt{1-\frac{{\nu }^2}{c^2}}$ for $\iota$ and $20y$ for $t$ into equation (1).

$\begin{array}{c}\nu =\frac{\left(20\text{cy}\right)\sqrt{1-\frac{{\nu }^2}{c^2}}}{20y}\\ =c\sqrt{1-\frac{{\nu }^2}{c^2}}\end{array}$

We further simplify the derived expression to obtain $\nu$ as follows:

$\begin{array}{l}\nu =c\sqrt{1-\frac{{\nu }^2}{c^2}}\\ \frac{\nu }{c}=\sqrt{1-\frac{{\nu }^2}{c^2}}\\ \frac{{\nu }^2}{c^2}=1-\frac{{\nu }^2}{c^2}\\ \frac{{\nu }^2}{c^2}+\frac{{\nu }^2}{c^2}=1\end{array}$

Solve further as:

$\begin{array}{l}\frac{2{\nu }^2}{c^2}=1\\ \frac{{\nu }^2}{c^2}=\frac{1}{2}\\ {\nu }^2=\frac{c^2}{2}\\ \nu =\frac{c}{\sqrt{2}}\end{array}$

Therefore, the value of speed of spaceship is$\nu =\frac{c}{\sqrt{2}}$.