Physics, asked by nkiitm, 11 months ago

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 x 10^27 kg as proton mass.​

Answers

Answered by CarliReifsteck
0

Given that,

Distance = 8.7 lye

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}

l'=\dfrac{25}{1.4}

l'=17.86\ m

The volume of spaceships is

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

The original volume of spaceships is

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

The mass of 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

Due to length transformation mass will be change

m'=\rho 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). The time takes your spaceship to reach Sirius in 12.4 years.

(b). The mass of interstellar gas is 1.195\times10^{27}\ kg

Similar questions