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Chapter 33: Problem 31

You wish to travel to a star \(N\) light years from Earth. How fast must you go if the one-way journey is to occupy \(N\) years of your life?

Short Answer

Expert verified
The required velocity \(v\) for the traveller's journey is, in units where the speed of light equals 1, \(v = \sqrt{1 - 1/N^2}\).

Step by step solution

01

Understand the Time Dilation

The time dilation formula in special relativity is \(T = T_0/ \sqrt{1 - v^2/c^2}\) where \(T\) is the dilated time, \(T_0\) is the time experienced by the observer in motion (in this case the individual travelling to the star), \(v\) is the velocity of the observer, and \(c\) is the speed of light.
02

Formulate the Equation for this Particular Journey

The distance to the star is \(N\) light years and the time experienced by the traveller in question is also \(N\). In terms of light years, the speed of light \(c\) equals 1, so the time dilation equation simplifies to: \(N = N/ \sqrt{1 - v^2}\). This is the equation which needs to be solved to find \(v\), the velocity required for the journey.
03

Solve for the Required Velocity

Rearrange the equation to isolate \(v\): in this case, square both sides of the equation to get \(N^2 = N^2/(1-v^2)\), simplify and solve for \(v\). This gives \(v^2 = 1 - 1/N^2\), and therefore \(v = \sqrt{1 - 1/N^2}\). This is the required velocity for the journey.

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Most popular questions from this chapter

The rest energy of an electron is \(511 \mathrm{keV}\). What's the approximate speed of an electron whose total energy is \(1 \mathrm{GeV} ?\) (Note: No calculations needed!)

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