Ask AI: Tom leaves on 1st Jan 2024, travels at 50% the speed of light, and returns to earth on 31st Dec 2024. How many earth years will he have aged?

Such a question can be resolved using the principle of time dilation in Einstein's theory of relativity, which states that time passes slower for a moving object compared to an object at rest. The formula for time dilation is t = t0 / sqrt(1 - v^2 / c^2), where t is the dilated time, t0 is the time in the rest frame, v is the velocity of the moving object, and c is the speed of light. Given that Tom has been travelling at 50% the speed of light, we plug the numbers into the formula: v would be 0.5c, so v^2 / c^2 is 0.25. This leaves us with sqrt(1 - 0.25) underneath, which is sqrt(0.75), or approximately 0.866. However, in this problem, we're told that he returns to Earth after a year, which means that that's the dilated time t (rather than the rest time t0). The time Tom has aged, then would require rearranging the formula to solve for t0 = t * sqrt(1 - v^2 / c^2). Substituting in the numbers t = 1 year and sqrt(1 - v^2 / c^2) = 0.866, we find that t0 is approximately 0.866 years, or about 10 and a half months. So, if he leaves earth on 1st Jan 2024 and returns on 31st Dec 2024, Tom will have aged about 10.5 months while Earth will have experienced a full year.