It is the year 2150 and the United Nations Space Federation has finally perfected the storage of antiprotons for use as fuel in a spaceship. (Antiprotons are the antiparticles of protons. We discuss antiprotons in Chapter 3.) Preparations are under way for a manned spacecraft visit to possible planets orbiting one of the three stars in the star system Alpha Centauri, some 4.30 lightyears away. Provisions are placed on board to allow a trip of 16 years’ total duration. How fast must the spacecraft travel if the provisions are to last? Neglect the period of acceleration, turnaround, and visiting times, because they are negligible compared with the actual travel time.
Strategy The time interval as measured by the astronauts on the spacecraft can be no longer than 16 years, because that is how long the provisions will last. However, from Earth we realize that the spacecraft will be moving at a high relative speed v to us, and that according to our clock in the stationary system K, the trip will last T = 2L/v, where L is the distance to the star.
Because provisions on board the spaceship will last for only 16 years, we let the proper time T_{0}^{\prime} \text { in system } K ^{\prime} be
16 years. Using the time dilation result, we determine the relationship between T, the time measured on Earth, and the proper time T_{0}^{\prime} to be
T=\frac{2 L}{v}=\frac{T_{0}^{\prime}}{\sqrt{1-v^{2} / c^{2}}} (2.20)
We then solve this equation for the required speed v.
