Question

You wish to make a round trip from Earth in a spaceship, traveling at constant speed in a straight line for exactly 6 months (as you measure the time interval) and then returning at the same constant speed. You wish further, on your return, to find Earth as it will be exactly 1000 years in the future. (a) To eight significant figures, at what speed parameter $\mathit{\beta }$must you travel? (b) Does it matter whether you travel in a straight line on your journey?

Short answer

(a) The speed parameter is $0.99999950$.

(b) It does not matter whether the journey is in straight line or not.

Step-by-step solution

Identification of given data

The duration of traveling in spaceship is $t_0=6$months$=0.5y$

The duration passed on Earth for future is $t=1000y$

The Lorentz factor is used to find the speed parameter which is the ratio of the speed of particle with speed of light.

Determination of speed parameter  (a) 

The speed parameter for travel is given as:

$\beta =\sqrt{1-{\left(\frac{2t_0}{t}\right)}^2}$

Substitute all the values in equation.

$$\begin{gathered} \beta =\sqrt{1-{\left(\frac{2\left(0.5y\right)}{0.99999950}\right)}^2} \\ \beta =0.99999950 \end{gathered}$$

Therefore, the speed parameter is 0.99999950 .

Effect of straight-line journey in travel  (b) 

The speed parameter is independent from the acceleration of spaceship so speed parameter does not change with change in path other than straight line. It does not matter that journey is in straight line or in any other path.