Question

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?

Answer 1

`v approx 0.75c`

`t approx 5.7 yr`

Explanation:

From the Lorentz transform:

`x = gamma (x' + vt')`

• `x' = 0`

• `t' = 3.8 y`

• `x = 4.3 \ c*y`

`v = x/(t' gamma) = x/(t') sqrt(1 - v^2/c^2)`

With `v/c = beta`, `implies beta = 4.3/3.8 sqrt(1 - beta^2)` which solves as `β≈0.75` so `v = 0.75c`

For the time transform:

`t = gamma (t' + (vx')/c^2) = (3.8)/sqrt(1 - 0.75^2) approx 5.7 yr`