Special relativity of a rocket

[vorcil]

1)
How fast must a rocket travel on a journey to and from a distant star so that the astronauts age 12.0 years while the Mission Control workers on Earth age 130 years ? c


2)
As measured by Mission Control, how far away is the distant star? in light years


my attempt
1)
Time in moving reference frame = (sqrt(1-beta))*time in inertial reference frame

12/130 = sqrt(1-beta)
12/130 ^2 = 1- beta
12/130^2 = Tn
tn = 1-beta
beta = 1-td
beta = v^2/c^2

converting light years to seconds (1ly = 31556296 seconds)
((12*31556926) / (130*31556926))^2 = 8.520*10^-3
1-(8.520*10^-3) = 0.99147 = beta
v^2/c^2 = 0.99147
sqrt(0.99147*c^2) = v
v/c = 0.9957 c which is 0.9957 as a fraction of the speed of light that the rocket has to be traveling
this was correct

2)
not quite sure how to solve the next one

 

[Doc Al]

my attempt
1)
Time in moving reference frame = (sqrt(1-beta))*time in inertial reference frame

12/130 = sqrt(1-beta)
12/130 ^2 = 1- beta
12/130^2 = Tn
tn = 1-beta
beta = 1-td
beta = v^2/c^2

converting light years to seconds (1ly = 31556296 seconds)
((12*31556926) / (130*31556926))^2 = 8.520*10^-3
1-(8.520*10^-3) = 0.99147 = beta
v^2/c^2 = 0.99147
sqrt(0.99147*c^2) = v
v/c = 0.9957 c which is 0.9957 as a fraction of the speed of light that the rocket has to be traveling
this was correct

Good. Note on terminology: Beta usually stands for v/c, not v^2/c^2.

2)
not quite sure how to solve the next one

According to Mission Control, how fast was the rocket moving and how long did it take to reach its destination? (Use basic kinematics.)

 

[nlightner]

, but here's my attempt:

The distance to the distant star can be calculated using the formula for distance in special relativity:

d = (v*t)/sqrt(1-(v^2/c^2))

We know that the time experienced by the astronauts on the rocket is 12 years, so t = 12. We also know that the speed of the rocket is 0.9957 c, so v = 0.9957 c. Plugging these values into the formula, we get:

d = (0.9957 c * 12 years)/sqrt(1-(0.9957 c)^2) = 12.020 light years

Therefore, the distance to the distant star, as measured by Mission Control, is approximately 12.020 light years.