Rocket in space travels between two stars

[henrco]

Hi,
Could I please get some guidance on if my approach and solution are correct here.
I feel I'm on the right track. But even though the answer to part b) feels right, it would be helpful to get advice.

Homework Statement

A space rocket travels between two stars separated by 10 light years. Observers on satellites orbiting the stars record a duration of 12 years for the journey.

a) What duration is recorded on the rocket’s clocks.
b) What is the separation of the stars in the rocket’s frame of reference? Explain your reasoning.

Homework Equations

v = d/t
ε = √1 - v^2/c^2

The Attempt at a Solution

Calculate the velocity of the space rocket.
Distance and time in light years and years, convert to meters and seconds.
v = d/t
d = 10 light years (1 light year = 9.46 x 10^15 m)
t = 12 years (1 year = 3.156 x 10^7 sec)

v = 10 (9.46 x 10^15 m) / 12(3.156 x 10^7) = 2.50 x 10^8 m/s

c (speed of light) = 2.998 x 10^8
Calculate ε = √1- (2.50x10^8)^2 / (2.998x10^8)^2 = 0.55

Calculate time recorded on internal ships clocks. = 0.55 x 12 = 6.62 years

Answer a) 6.62 years

Part b)
d = t.v = (6.62)(3.156x10^7) x (2.5x10^8) = 5.22318 x 10^16 m
d = 8 light years

Answer b) 8 light years

 

[PeroK]

Hi,
Could I please get some guidance on if my approach and solution are correct here.
I feel I'm on the right track. But even though the answer to part b) feels right, it would be helpful to get advice.

Homework Statement

A space rocket travels between two stars separated by 10 light years. Observers on satellites orbiting the stars record a duration of 12 years for the journey.

a) What duration is recorded on the rocket’s clocks.
b) What is the separation of the stars in the rocket’s frame of reference? Explain your reasoning.

Homework Equations

v = d/t
ε = √1 - v^2/c^2

The Attempt at a Solution

Calculate the velocity of the space rocket.
Distance and time in light years and years, convert to meters and seconds.
v = d/t
d = 10 light years (1 light year = 9.46 x 10^15 m)
t = 12 years (1 year = 3.156 x 10^7 sec)

v = 10 (9.46 x 10^15 m) / 12(3.156 x 10^7) = 2.50 x 10^8 m/s

c (speed of light) = 2.998 x 10^8
Calculate ε = √1- (2.50x10^8)^2 / (2.998x10^8)^2 = 0.55

Calculate time recorded on internal ships clocks. = 0.55 x 12 = 6.62 years

Answer a) 6.62 years

Part b)
d = t.v = (6.62)(3.156x10^7) x (2.5x10^8) = 5.22318 x 10^16 m
d = 8 light years

Answer b) 8 light years

Converting to m/s is horrible! Why not stick with light years, years and velocity as a fraction of $c$?

The answer to a) looks correct, but b) is wrong. You can see this immediatley by noting that it cannot take 6.62 years for the star to travel 8 light years relative to the rocket.

 

[henrco]

Thanks for your reply.

However I've been looking at part b) which you said is wrong and I can't understand how to derive the correct solution.
b) What is the separation of the stars in the rocket’s frame of reference? Explain your reasoning.

If the duration on the rockets clocks is 6.62 years, answer to part a. The I would expect the distance from the rockets
frame of reference to be shorter than 10 light years (the external reference frame). Is my reasoning right or wrong?

If I right or wrong I'd appreciate a push in the right direction as I'm stuck on this.

 

[SammyS]

Thanks for your reply.

However I've been looking at part b) which you said is wrong and I can't understand how to derive the correct solution.
b) What is the separation of the stars in the rocket’s frame of reference? Explain your reasoning.

If the duration on the rockets clocks is 6.62 years, answer to part a. The I would expect the distance from the rocket's
frame of reference to be shorter than 10 light years (the external reference frame). Is my reasoning right or wrong?

If I right or wrong I'd appreciate a push in the right direction as I'm stuck on this.

Indeed, the distance that the star travels in the rockets frame must be less than 10 light-years.

How does PeroK know that 8 light-years is incorrect?

How far does light travel in 6.62 years?

The relative speed of the star to the rocket ship is less than the speed of light, so the star travels an even shorter distance in 6.62 years.​

 

[BvU]

A space rocket travels between two stars separated by 10 light years. Observers on satellites orbiting the stars record a duration of 12 years for the journey.

My impression from this statement is that there are two stars and the rocket travels from one star to the other. However, in that case I have difficulty interpreting the observer recordings. What do they represent ? The time between such an observer seeing the rocket at star 1 and seeing the rocket at star 2 I would presume ?

 

[henrco]

Thanks for your reply SammyS but the 'penny' hasn't dropped for me yet.

Indeed, the distance that the star travels in the rockets frame must be less than 10 light-years.

How does PeroK know that 8 light-years is incorrect?
How far does light travel in 6.62 years?
The relative speed of the star to the rocket ship is less than the speed of light, so the star travels an even shorter distance in 6.62 years.

To answer your question, light would travel 6.62 light years in 6.62 years.
However the speed of of the rocket is 83.39% speed of light. So I'd expect the distance to be larger than 6.62 light years but certainly less than 10.
However you've indicated that the distance should be even shorter than 6.62 light years.

I don't understand how this could be, could you please explain.

 

[SammyS]

Thanks for your reply SammyS but the 'penny' hasn't dropped for me yet.

To answer your question, light would travel 6.62 light years in 6.62 years.
However the speed of of the rocket is 83.39% speed of light. So I'd expect the distance to be larger than 6.62 light years but certainly less than 10.
However you've indicated that the distance should be even shorter than 6.62 light years.

I don't understand how this could be, could you please explain.

The speed of light is $\displaystyle \ c=1\frac{\mathtt{light\ year}}{\mathtt{year}} \ .\$\ Right?

So, in 6.62 years, light travels 6.62 light-years. The rocket is slower than the speed of light, so the distance it travels is less than 6.62 light-years.

 

[henrco]

Thank you for this. Clearly my brain decided not to work earlier...

Since the rocket travels at 83.39% of the speed of light, or 83.39c
Then the distance traveled will be 6.62 x 83.39/100 = 5.52 light years.