Question

Because you are moving with an enormous speed, your mission from the previous problem A.1 will be influenced by the eects of time dilation described by special relativity: Your spaceship launches in June 2020 and returns back to Earth directly aer arriving at Sirius. (a) How many years will have passed from your perspective? (b) At which Earth date (year and month) will you arrive back to Earth?

Answer 1

Given that,

The spaceship launches in June 2020 and returns back to Earth directly aer arriving at Sirius.

Suppose, You are on an interstellar mission from the Earth to the 8.7 light-years distant star Sirius. Your spaceship can travel with 70% the speed of light and has a cylindrical shape with a diameter of 6 m at the front surface and a length of 25 m. You have to cross the interstellar medium with an approximated density of 1 hydrogen atom/m³ . (A) Calculate the time it takes your spaceship to reach Sirius.

Given that,

Distance = 8.7 ly

Speed = 70%

Diameter = 6 m

Length = 25 m

(A). We need to calculate the time

Using formula of speed

$v=\dfrac{d}{t}$

$t=\dfrac{d}{v}$

Where, d = distance

v = velocity

Put the value into the formula

$t=\dfrac{8.7\times9.46\times10^{15}}{3\times10^{8}\times0.70}$

$t=391914285.714\ sec$

$t=\dfrac{39.2\times10^{7}}{3.15\times10^{7}}$

$t=12.4\ years$

We need to calculate the prospective time

Using formula of time dilation

$t=\gamma t'$

$t=\dfrac{t'}{\sqrt{1-\dfrac{v^2}{c^2}}}$

$t'=t\times\sqrt{1-\dfrac{v^2}{c^2}}$

Put the value into the formula

$t'=12.4\times\sqrt{1-\dfrac{0.7^2c^2}{c^2}}$

$t'=12.4\times\sqrt{1-0.7^2}$

$t'=8.8\ years$

$t'=9\ years$

We need to calculate the value of lorentz factor

Using formula of lorentz factor

$\gamma=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$

Put the value into the formula

$\gamma=\dfrac{1}{\sqrt{1-\dfrac{(0.7c)^2}{c^2}}}$

$\gamma=\dfrac{1}{\sqrt{1-(0.7)^2}}$

$\gamma=1.4$

(a). The spaceship takes time to reach the Sirius is

$t'=9\ years$...(I)

Suppose, the spaceship directly back to earth when it reach the sirius.

So, the spaceship takes time to back the earth is

$t''=9\ years$...(II)

We need to calculate the time on our prospective

The total time in whole journey

$t=t'+t''$

Put the value into the formula

$t=9+9$

$t=18\ years$

(b). The spaceship starts journey at June 2020.

$t_{s}=June\ 2020$

We need to calculate the time when the spaceship arrive back to earth

Using formula for total time

$T=t_{s}+t$

Put the value into the formula

$T=June\ 2020+18\ years$

$T=June\ 2038$

Hence, (a). The spaceship passed from our prospective in 18 years.

(b). The spaceship arrive back to earth is June 2038.