Question

Arthur Clark’s The Wind from the Sun (1963) describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of 0.001g = 0.0098m/s² .Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of the speed c = 3×10⁸m/s of light. How long will it take the spacecraft to catch up with the projectile, and how far will it have traveled by then? 

Answer 1

Set t_f to be the time at which the spacecraft catches up with the projectile. The total distance traveled by the spaceship will be: 

The total distance traveled 

by the projectile will be:

x₂(t_f ) = vt_f .

When the spaceship catches up with the projectile, the distances are the same, and we have:

vt_f = 1 2 at²_f . 

Solving this for tf we get:

So, about 194 years! The total distance traveled in that time will be:

vt_f ≈ 1.837 × 10¹⁷m.

This is about 19.4 light years.