Answer to Question #153990 in Mechanics | Relativity for Tejpal

Question

Prime takes a trip in spaceship to a distant station (distance L = 1 light year) at uniform speed (v=0.8c). 

After reaching the distant station, Prime takes a return spaceship to meet Unprime  in Ahmedabad. Explain who is younger after this journey: Prime or Unprime, and  by how many years? Use units of distance as light year and that of time as year.

Answer 1

Explanations & Calculations

• Time dilation/expansion is the base theory applied here.
• And what this question describes is the thought called "twin paradox".
• The earthbound Unprime sees that Prime takes a longer time to return to earth equally Prime sees that Unprime has aged more as he returns.
• The relationship for the time dilation is given by,

$\qquad\qquad \begin{aligned} \small \Delta T&= \small \gamma\Delta \tau \end{aligned}$ $\qquad\qquad \begin{aligned} \small \tau&= \small \text{Lorentz factor}=\sqrt{1-\frac{v^2}{c^2}}\\ \small \Delta T&= \small \text{duration relative to earthbound one} \\ \small \Delta \tau &= \small \text{duration relative spacebound one} \end{aligned}$

• As seen by Prime, it takes $\small 2\times \frac{1c\times year}{0.8c}=2.5 years$ for the whole trip.
• Therefore, this time is seen as

$\qquad\qquad \begin{aligned} \small \Delta t&= \small \frac{1}{\sqrt{1-\frac{(0.8c)^2}{c^2}}}\times 2.5\text{years}\\ &=\small 4.167 \text{years} \end{aligned}$

by Unprime.

• Therefore, Prime is 4.167 years younger than Unprime.