How fast must a spaceship travel to reach Alpha Centauri in 10 years?

[spXq]

The distance to Alpha Centauri is 4,3 light years. How fast would a spaceship have to travel to get there in 10 years, according to the crew?

The answer *should* be 0,395c. So far I've gotten all sorts of answers but not much close, so I seem to be approaching the problem the wrong way.

 

[Doc Al]

Show what you've done so far.

Hints: What's the basic idea of time dilation? What's the definition of velocity?

 

[spXq]

The first part of the assignment asks for the time required when traveling at 30% of light speed. Given a gamma factor of 1,048 I get ~13,7 years, which is correct (so the gamma factor must be correct as well).

velocity = distance / time

But...

for t = 10 and d = 4,3

v = 4,3 / 10
v = 0,43, which is wrong

taking length contraction into account gives

4,3/1,048 = 4,1

v = 4,1 / 10
v = 0,41, which is still wrong

 

[Doc Al]

Edit: velocity = distance / time

Good. Keep going.

From Earth's viewpoint, you have the distance. What's the time?

 

[spXq]

Good. Keep going.

From Earth's viewpoint, you have the distance. What's the time?

Hmm... i don't know. :/

The distance is 4,3 ly but I don't know the speed, so how do I find the time?

 

[spXq]

Hang on... the time for the crew is 10 years so the time for the observer is t/gamma

10/1,048 = 9,54 years

Edit: no that's not right... I have no idea what I'm doing

Edit2: the time from Earth's viewpoint is d * gamma = 4,3 * 1,048 = 10,48 years, while the time is 10 years for the crew

 

[Doc Al]

Hang on... the time for the crew is 10 years so the time for the observer is t/gamma

You have that reversed. If the ship time is 10 years, then to Earth observers it will be longer: t*gamma, not t/gamma.

10/1,048 = 9,54 years

In addition to what I already pointed out, do not use the gamma from the previous part of the question.

So continue with that velocity equation, v = d/t_earth.