Time to travel at relativistic speed

[jl1642]

I have a relatively simple problem that I'm having trouble with.
A ship is going at 0.90 c, over a distance of 80 light years. In my text the method for determining the time spent traveling is:

80 years + (0.1 * 80) = 88 years
This method seems logical, at 0.9 of the speed of light this trip should take 80 years plus that missing 0.1 of the speed of light.

I tried t = d/v :
t = 80 years / (0.9 light years per year)
t = 88.888889

I'm not sure what is going on, I see no logical error with either approach. Any help is appreciated.

 

[Doc Al]

80 years + (0.1 * 80) = 88 years
This method seems logical, at 0.9 of the speed of light this trip should take 80 years plus that missing 0.1 of the speed of light.

This does not make sense to me. This was in your textbook?

You are going slower than lightspeed by a factor of 1/.9 = 1.111..., not 1.10. So it should take you 80 years + 0.111...*80 = 88.88...

I tried t = d/v :
t = 80 years / (0.9 light years per year)
t = 88.888889

This makes sense. (Note that it's equivalent to what I said above.)

 

[jl1642]

Thanks! I thought I was right, it just messed me up that the textbook was wrong.

 

[vela]

Did your textbook say it was equal to 88 years or approximately equal to 88 years?

Recall the sum of a geometric series:

$$1+x+x^2+\hdots = \frac{1}{1-x}$$

If x is small, you can use the approximation

$$\frac{1}{1-x}\cong 1+x$$

where you truncate the series after one term without introducing much error. That may be what your book was doing. If it says the time was exactly 88 years, though, it's wrong.

 

[jl1642]

No, the textbook introduces the method with an example question, exactly as I wrote above, with 80 + (0.1*80).