TPM Idea Lab: The World’s Smallest Electric Motor Is Made From a Single Molecule “As for uses–oh, yeah, there are actual uses for this thing!–scientists believe that this molecule could further be developed into a motor to power nano-sized devices in fields such as medicine.” [Very very small cars]

The speed of travel is increasing over time. While we might argue about the rate at which speed is increasing, this long term observation is consistent with improving transportation technology.
Suppose we model technological progress so that the speed of travel is increasing with a compound interest model
V_t = V_{t-1}*\alpha
Where \alpha > 1. (E.g. if speed were increasing at 1 percent compounded annually, alpha would be 1.01). Escape velocity at the surface of the earth is 11.2 km/s.
According to Wikipedia “[Robert H.] Goddard and his team launched 34 rockets between 1926 and 1941, achieving altitudes as high as 2.6 km (1.62 miles) and speeds as high as 885 km/h (550 mph).” (246 m/s).
In 1959, A Rocket by the Soviets hit the moon at 7,500 mph (3,352.8 m / s). It traveled 236,875 miles in about 35 hours, for an average speed of 3,025.5 m / s.
In 2006, an Atlas V rocket launched toward Pluto is traveling at 36,000 MPH (16,093.44 m / s), taking 9 years (including a Jupiter slingshot).
Proxima Centauri is 4.2 light years (~ 4.0 × 10^13 kilometers) from Earth, or 270,000 times more distant than the Sun.
If we are currently able to travel at 16,000 m/s it would take 2.5x 10^12 seconds to reach Alpha Centauri (about 79000 years).
However, we have maximum speed growing at a compound rate of about 3.39 percent annually (rising from 3,025 in 1959 to 16,093 in 2006). (\alpha ~ 1.0339) (In fact the rate was higher from Goddard to the Russians, and so may be dropping over time, but let’s assume this rate is fixed).
So we want to be able travel to Proxima Centauri in less than 79000 years, if we can maintain this rate of speed growth for 100 years, our speed would be 448,718 m/s, but if we wait 1000 years, our speed would be 4,815,654,367,278,682,000 m/s. Since the speed of light = 299,792,458 m / s, that is our upper limit (if we believe Einstein rather than Roddenberry). It turns out, that at 3.39 percent annual compound increase in speed, we reach a maximum speed of 298,736,235 m/s in 295 years from 2006, or the year 2301, and reach the speed of light exactly shortly thereafter (Tuesday afternoon, 5:53 pm).
Hence, we should leave for the stars in 2301 to minimize travel time, reach them in 2305, and try to avoid destroying the Earth and/or Solar System before then. That will get us there far sooner than trying to leave much earlier.
We could leave as early as 2240 and arrive in 2275, but if we wait until 2251, we still arrive in 2275. We save travel time each year we wait after that, but will arrive a bit later.

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