# Re: Mars in two weeks?

From: hal@finney.org
Date: Fri Jan 05 2001 - 12:32:54 MST

xgl writes:
> at 1 g, the velocity gets relativistic pretty fast. if my math
> hasn't failed me, by the time the craft is halfway to mars, it'd be doing
> about 0.4 c -- decelerating at 1 g as well, it will get from earth to mars
> in a little less than 3 days.

This is not quite right. Earth-Mars distance is about 60 million miles
at closest approach. 1 g is 32 ft/sec^2, or .006 mi/sec^2. 1/2 a t^2
for 30 million miles is about 100,000 seconds or 27.6 hours to turnaround
point, for a total of 2 days, 7 hours (so you were right about the time
estimate).

However the velocity will be a*t, or .006 * 100,000 which is about 600
miles per second, far from the speed of light which is 186,000 mps.

(Apologies for all the English units but for an old fashioned guy like
me it is more intuitive.)

There is an interesting way to see that the speed estimate has to
be wrong. In relatively theory we sometimes use "geometrical units",
which sets the speed of light c to be the dimensionless quantity 1.
In these units, speeds are dimensionless fractions, and therefore time
and space have the same units. A light-year is the same as a year,
in this system. (You can also bring mass into the system by treating
a given mass as equivalent to the size of the black hole it would form.)

In dimensionless units, 1 g is 9.8 m/s^2 * (1 s / 3E8 m) or about one 30
millionth per second, or about one per year. In the simple approximation
v = a * t so given that g is 1/year it will take about a year at that
accelration to reach a speed of approximately 1, which is the speed
of light.

So if you know that Earth's gravitational acceleration is about 1/year
in these units (easy to remember) you know right away that you won't
be close to the speed of light after only a day or two of acceleration,
which is how I know that the speed figure was wrong.

BTW the fact that Earth's gravitational acceleration is approximately
equal to its orbital period is pure coincidence AFAIK and is not true
of any of the other planets, probably.

Hal

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