How long to almost reach light speed?

Hold a rocket at a steady felt acceleration and watch the Solar System shrink behind it. At 1 g, about one onboard year reaches 76% of c, already half a light-year from Earth.

v/c = tanh(ατ/c)constant proper acceleration

Inner Solar System · top viewView radius 40 AU

Felt acceleration α0.1–10 g

0.1 g10 g

Target speed vnever reaches c

10% c99.999996% c

Dry mass mexcludes fuel

1 kg100 t

Target speed reached after

Minimum kinetic energy

Try

A massive rocket never reaches light speed. Constant proper acceleration is the acceleration passengers feel. Starting from rest, v = c tanh(ατ/c), Earth-frame distance x = c²[cosh(ατ/c) − 1]/α, and kinetic energy E_k = (γ − 1)mc². The star directions are schematic; radial distances are to scale. Energy is for the selected dry mass only and excludes propellant, exhaust, inefficiency and braking. Sources: Wikipedia on hyperbolic motion, relativistic kinetic energy, the heliosphere, Oort Cloud Alpha Centauri and Voyager 1.