Deep Seek, how long would a 2 G acceleration get you near light speed

DeepSeek, a LLM that I love, gave me the answer to a nagging question, if we could accelerate constantly at 2G how long would it take to get near light speed, here is the answer:

To determine how long it would take for a spaceship accelerating at 2 G (where 1 G = 9.81 m/s²) to reach near light speed (c ≈ 299,792,458 m/s) as observed from Earth, we can use the principles of relativistic motion under constant proper acceleration.

Key Equations:

1. Proper Acceleration (a): The acceleration experienced by the spaceship's crew (2 G = 19.62 m/s²).

2. Relativistic Velocity (v) as a function of time (t) in Earth's frame:

   $$v(t)=\frac{at}{\sqrt{1+{\left(\frac{at}{c}\right)}^2}}$$

   • As $t$ increases, $v$ approaches $c$ but never exceeds it.

3. Time to reach a significant fraction of $c$:

   • The spaceship's velocity approaches $c$ asymptotically, so we define "near light speed" as some high fraction (e.g., 0.9c, 0.99c, etc.).

   • The time to reach a given fraction $f$ (where $v=fc$) is:

     $$t=\frac{fc}{a\sqrt{1-f^2}}$$

Calculation for 2 G Acceleration:

Given:

• $a=2\text{G}=19.62{\text{m/s}}^2$

• $c=299,792,458\text{m/s}$

Example: Time to reach 0.9c:

$$t=\frac{0.9\times 299,792,458}{19.62\times \sqrt{1-{0.9}^2}}=\frac{269,813,212}{19.62\times 0.4359}\approx 31,600,000\text{seconds}\approx \ast \ast 1year\ast \ast$$

Time to reach 0.99c:

$$t=\frac{0.99\times 299,792,458}{19.62\times \sqrt{1-{0.99}^2}}=\frac{296,794,533}{19.62\times 0.141}\approx 107,000,000\text{seconds}\approx \ast \ast 3.4years\ast \ast$$

Time to reach 0.999c:

$$t\approx \frac{0.999\times c}{19.62\times 0.0447}\approx 10.7\text{years}$$

Observations:

• Due to relativistic effects, the spaceship never actually reaches $c$, but it can get arbitrarily close.

• From Earth's perspective, the ship's acceleration appears to decrease as it approaches $c$, even though the crew always feels a constant 2 G acceleration.

• After about 1 year at 2 G, the ship is moving at ~0.9c.

• After 3.4 years, it reaches ~0.99c.

• After 10.7 years, it reaches ~0.999c.