# Special Relativity (1 Viewer)

#### Khan.Paki

##### Member

The distance to the star X is 4.367 light years as measured from Earth. Using a relevant calculation, explain how a rocket could complete the journey from Earth to star X in 3.28 years.

#### Drsoccerball

##### Well-Known Member

The distance to the star X is 4.367 light years as measured from Earth. Using a relevant calculation, explain how a rocket could complete the journey from Earth to star X in 3.28 years.
As you approach relativistic speeds the distance required to travel will dilate and according to Earth the time taken will also dilate, thus making the trip 3.28 years

#### Khan.Paki

##### Member
Is that all? It says explain with calculations.

#### rand_althor

##### Active Member
$Using the formula for velocity, where d is contracted length observed by a person on the rocket, and t is the proper time observed by a person on the rocket:$
\begin{align*}v&=\frac{d}{t} \\v&=\frac{4.367\sqrt{1-\frac{v^2}{c^2}}}{3.28} \\\frac{3.28}{4.367}v&=\sqrt{1-\frac{v^2}{c^2}} \\(0.75v)^2&=1-\frac{v^2}{c^2} \\\frac{v^2}{c^2}+0.56v^2&=1 \\ \textup{Let }c&=1\\v^2+0.56v^2&=1 \\1.56v^2&=1 \\v^2&=\frac{1}{1.56} \\v&=0.80c \\\end{align*}

\begin{align*}\textup{If }v&=0.80c \\v&=\frac{d}{t} \\t&=\frac{d}{v} \\&=\frac{l_o\sqrt{1-\frac{v^2}{c^2}}}{0.80} \\&=\frac{4.367\sqrt{1-0.80^2}}{0.80} \\&=\frac{2.62}{0.80} \\&=3.28 \textup{ years} \end{align*}

If the rocket is moving at speed of 0.80c, it will be able to travel 4.367 light years in 3.28 years. This is because from an observer on Earth's perspective, the distance the rocket is traveling will contract.

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