# The nearest star to Earth is Proxima Centauri, 4.3 light-years away. At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 3.8 years, as measured by travelers on the spacecraft?

## How long does the trip take according to Earth observers?

Apr 12, 2018

$v \approx 0.75 c$

$t \approx 5.7 y r$

#### Explanation:

From the Lorentz transform:

$x = \gamma \left(x ' + v t '\right)$

• $x ' = 0$

• $t ' = 3.8 y$

• $x = 4.3 \setminus c \cdot y$

$v = \frac{x}{t ' \gamma} = \frac{x}{t '} \sqrt{1 - {v}^{2} / {c}^{2}}$

With $\frac{v}{c} = \beta$, $\implies \beta = \frac{4.3}{3.8} \sqrt{1 - {\beta}^{2}}$ which solves as β≈0.75 so $v = 0.75 c$

For the time transform:

$t = \gamma \left(t ' + \frac{v x '}{c} ^ 2\right) = \frac{3.8}{\sqrt{1 - {0.75}^{2}}} \approx 5.7 y r$