# Most Effiecent Relativistic Speed (for people back home)

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#### chebby

##### Guest
Hi,

I was thinking if we ever will get to travel at speeds close to speed of light, what would be the most efficent speed (from the point of view of earth as they want you to return as soon as possible)?

For example let's say we are traveling to a star close 4 light years away. To make it easy let's discard acceleration (it makes calculations harder) and use a scenario when you leave and arrive with the same speed (I know it's unrealistic but to keep things simple). To do this I made a table listed below that used Relativistic Factor I got from the relativistic calculator from http://www.1728.com/reltivty.htm

From the point of view of people in the spaceship it is easy - the closer speed is to 1.0 the better. Therefore going at .999 is the best, it gets people to the star in 4.04 years. However, a whooping 90.4 years passes on earth. Not good.

Let's try a speed of .1, then 40.4 years will pass on earth, also no good.

By trying different values I found .78 to give the smallest time (8.2) years.

I think a differential equaltion can be set up to find the most efficient speed, but I it's been a while since my calculus classes so I would prabobly need some help. ALso, maybe somebody already figured out the formula for this and I am just not aware?

Code:
``````<pre>
********************************************************************************************************************************************
distance (light years)   speed(in C)     Rfactor		SpaceshipTime(T(years)=d/speed)	Earth TIme(T(years)=Tspaceship * RF)

4			.1		  1.01			40.0					40.4
4			.5		  1.15			8.00					9.2
4			.78		  1.60			5.13					8.2
4			.8		  1.67			5.00					8.35
4			.95		   3.20			4.21					13.472
4			.999		   22.4			4.04					90.495
********************************************************************************************************************************************
</pre>``````

The second part of my question is about fuel efficiency (I know it grows by the same factor.) Would going at the most time efficient speed also be the most efficient speed for fuel expenditure?

thanks,
Chebby

S

#### Shpaget

##### Guest
chebby":26dug0to said:
From the point of view of people in the spaceship it is easy - the closer speed is to 1.0 the better. Therefore going at .999 is the best, it gets people to the star in 4.04 years. However, a whooping 90.4 years passes on earth. Not good.
For the people in the ship a 4 ly trip traveling at 0,999c would seem like 0,179 years.
For the people on Earth the mission would last 4,004 years.

For 0,99999 c
For the people in the ship a 4 ly trip traveling at 0,99999c would seem like 0,0179 years.
For the people on Earth the mission would last 4,00004 years.

You see, the faster you go, the better for all.

The faster you go, more fuel you must consume. At non relativistic speeds it is twice the speed twice the fuel consumption (not calculating the mass of the fuel itself), but when you get to relativistic speeds, it grows exponentially since the mass of entire ship starts to grow.

C

#### chebby

##### Guest
Shpaget, thank you!

The possibility that my understanding of the phenomena was flawed had not crossed my mind.

This suddenly makes much more sense :mrgreen:

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