The nearest star to Earth is Proxima Centauri, 4.3 light-years away.

(a) At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 5.0 years, as measured by travelers on the spacecraft?

(b) How long does the trip take according to Earth observers?

a) v = 2,152 10⁸ m / s b) t = 2.71 10⁸ s or t = 85.93 year

Explanation:

a) In this special relativity exercise we have that time is measured in the same ship, so it is the proper time,

v = d / t

Let's reduce the distance to the SI system

d = 4.3 l and (9.46 1015 m / 1ly) = 40.678 10¹⁵ m

t = 5.0 y (365 day / 1 y) (24 h / 1 day) (3600s / 1h) = 1.89 10⁸ s

Let's calculate

v = 40.678 10¹⁵ / 1.89 10⁸

v = 2,152 10⁸ m / s

b) The time seen from the ground for which the ship moves is given by

t = t₀ / √ (1 - (v/c) ²)

Let's calculate

t = 1.89 10⁸ / √ (1 - (2.152 / 2.998) ²)

t = 1.89 10⁸ / 0.6962

t = 2.71 10⁸ s

Let's reduce this time to years

t = 2.71 10⁸ s (1h / 3600s) (1day / 24h) (1 and / 365 d)

t = 85.93 year