Doppler shift and number of signals in relativistic travel

[Delzac]

Homework Statement

An astronaut makes a round trip at 0.80 c to a star 20 light-years away, while
her husband remains on Earth. Each sends the other a signal once a year by
his or her own reckoning. Assume that the period of acceleration when the
astronaut turns to head back home is small and both survive the trip.

How many signals does the astronaut receive from her husband on the
outward trip?

Homework Equations

$$L \gamma = L_0$$

The Attempt at a Solution

I didn't use the length contraction formula even though the suggested solution uses it.
This is what i thought. As a observer on Earth i will see the astronaut traveling at .8c, i see my signal traveling at 1c, thus my signal gains on the astronaut .2c light year every year.

Then my simple arithmetic, i calculate the signal will reach the astronaut 4 times.

During the first year - .8 c light years covered
until the 4th year - 3.2c light years covered

On the 4th year, if i were to send a signal, it will take 16 years to get to him (light gains on him .2c ly per yr). So i calculated or assumed that the answer is 4 signals.

But the suggested answer is 5 and uses length contraction formula and Doppler Shift Formula. Why the difference?

Any help will be great.

 

[eczeno]

it is not clear what the question is asking you to find.

 

[Delzac]

My bad edited.

 

[D H]

The problem as stated suffers from what is called in computer science a fencepost problem. Here's a nice description of the problem: http://betterexplained.com/articles/learning-how-to-count-avoiding-the-fencepost-problem/.

One could reasonably argue that the correct answer is any of four, five, or six. Rephrasing the question so as to avoid the fencepost problem, let's say that the two send signals to one another frequently. The Earth-bound husband prefaces each his signals with "It's been <time span> since you left, Alice". What is the preface of the signal Alice receives when she reaches the turnaround point?

With this, the answer is five years.

 

[eczeno]

hi DH,

I read that article, what a treasure. and now I have just finished the article on sine, it is truly beautiful and lucid writing on mathematics; a rare treat.

cheers

 

[Delzac]

Got it! Thanks. Was careless.