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Chapter 20: Problem 26

You wish to make a round trip from Earth in a spaceship, traveling at constant speed in a straight line for 6 months and then returning at the same constant speed. You wish further, on your return, to find the Earth as it will be 1000 years in the future. \((a)\) How fast must you travel? \((b)\) Does it matter whether or not you travel in a straight line on your journey? If, for example, you traveled in a circle for 1 year, would you still find that 1000 years had elapsed by Earth clocks when you returned?

Short Answer

Expert verified
The spaceship must travel at approximately 99.99995% the speed of light to find the Earth as it will be 1000 years in the future upon returning from a 1-year round trip. Yes, the same amount of time would elapse on Earth regardless of the path taken, as long as the speed is constant and the journey lasts for 1 year from the spaceship's frame.

Step by step solution

01

Understanding Time Dilation

The time dilation formula from the theory of special relativity is given by: \(t' = t \sqrt{1 - \frac{v^2}{c^2}}\), where \(t'\) is the time experienced by the moving observer (in the spaceship), \(t\) is the time experienced by stationary observer (on Earth), \(v\) is the speed of the spaceship and \(c\) is the speed of light. We have to solve for \(v\), the speed of the spaceship.
02

Applying the Time Dilation formula

Given that \(t' = 1\) year (6 months out and 6 months back) and \(t = 1000\) years. Substituting these values and the speed of light \(c = 3 \times 10^8\) m/s into the time dilation formula gives us: \(1 = 1000 \sqrt{1 - \frac{v^2}{(3 \times 10^8)^2}}\). Then square both sides of the equation to get rid of the square root.
03

Solve for the speed \(v\)

After squaring, the equation becomes: \(1 = 1000000 \times (1 - \frac{v^2}{(3 \times 10^8)^2})\), which simplifies to: \(v^2 = (1 - \frac{1}{10^6}) \times (3 \times 10^8)^2\). Solving for \(v\), we find \(v \approx 0.9999995c\), the spaceship must travel at approximately 99.99995% the speed of light.
04

Address the circular path

The shape of the trip doesn’t matter, because the spaceship is moving at a constant speed throughout the journey, whether it's in a straight line or a circular path. The effect of time dilation will simply accumulate over the total time of travel, and the same 1000 years would have elapsed on Earth clocks when the spaceship returned.

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