Question

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. (b) Determine the mass of interstellar gas that collides with your spaceship during the mission. Note: Use 1.673 × 10−27 kg as proton mass.

Answer 1

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 time dilation

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

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

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

$\gamma=1.4$

(b). According to relativity,

Suppose the spaceship moves along x- axis

The spaceship's length will be change

$l'=\dfrac{l}{\gamma}$

Put the value into the formula

$l'=\dfrac{25}{1.4}$

$l'=17.86\ m$

The volume of spaceship is

$V'=\pi\dfrac{D^2}{4}\times l'$

The original volume of spaceship is

$V=\pi\dfrac{D^2}{4}\times l$

The density of interstellar medium is

$\rho=1\ hydrogen\ atom/m^3$

The mass of 1 hydrogen atom is

$m=1.673\times10^{-27}\ kg$

The original mass is

$m=\rho V$

We need to calculate the mass of interstellar gas that collides with your spaceship during the mission

Due to length transformation mass will be change

$m'=\rho V'$

Put the value of V'

$m'=\rho\times\pi\times\dfrac{D^2}{4}\times l'$

Put the value of l'

$m'=\rho\times\pi\times\dfrac{D^2}{4}\times \dfrac{l}{\gamma}$

$m'=\dfrac{m}{\gamma}$

Put the value into the formula

$m'=\dfrac{1.673\times10^{-27}}{1.4}$

$m'=1.195\times10^{-27}\ kg$

Hence, (a).your spaceship takes the time to reach Sirius is 12.4 years.

(b). The mass of interstellar gas that collides with your spaceship during the mission is $1.195\times10^{-27}\ kg$