In the age of interstellar travel, an expedition is mounted to an interesting star 2000.0 light-years from Earth. To make it possible to get volunteers for the expedition, the planners guarantee that the round trip to the star will take no more than \(10.000 \%\) of a normal human lifetime. (At that time the normal human lifetime is 400.00 years.) What is the minimum speed the ship carrying the expedition must travel?

The concept of a light-year is foundational when discussing interstellar travel. A light-year is the distance that light travels in a vacuum in one year, which equates to approximately 9.461 × 10^12 kilometers. To put that in perspective, it's about 63,241 times the distance between the Earth and the Sun.

When working with interstellar distances, converting light-years to a more familiar unit like kilometers helps us to understand the staggering scales involved in space travel. In our problem, the star is 2000 light-years away, meaning the round-trip covers 4000 light-years. By converting this distance into kilometers, we can then proceed to calculate meaningful values for the journey, like the speed of a spaceship.

So, to convert a light-year to kilometers, we use the formula: Distance in km = Distance in light-years × 9.461 × 10^12 km/light-year. Through this equation, we can bridge the gap between astronomical measurements and more conventional units of distance that relate to our everyday experiences.

When delving into the realm of interstellar travel, one cannot overlook the fascinating effect of time dilation as predicted by Einstein's theory of relativity. Time dilation is a consequence of the relative nature of time, which is affected by speed. As an object moves closer to the speed of light, time slows down for it relative to an observer at rest.

This becomes significant when discussing space travel at relativistic speeds, meaning speeds that are a significant fraction of the speed of light. For the brave travelers journeying to a distant star, time aboard the spacecraft would not pass at the same rate as it does for those staying on Earth. The faster the spacecraft travels, the more pronounced this time dilation effect becomes.

Understanding time dilation is crucial for planning such expeditions because the objective is to minimize the time experienced by the crew, ensuring they age much less during the journey than they would if they remained on Earth. For a spacecraft to achieve this, it must travel at a significant fraction of the speed of light, leading to relativistic effects that become a central factor in calculating the required travel velocity.

The term 'relativistic speed' refers to velocities that approach the speed of light, resulting in the need to consider Einstein's Special Theory of Relativity in our calculations. As we discuss interstellar travel, calculating the minimum speed for a spacecraft requires not only physics but an understanding of these relativistic effects.

In our textbook problem, we are aiming to solve for the minimum speed the ship must travel. We already established the round-trip distance in kilometers and the maximum time allowed in seconds. The final step is to use the formula: Minimum speed = Distance / Time, bearing in mind that the speed obtained must be practical within the framework of relativity.

This implies considering the maximum time dilated lifespan of the astronauts, a product of the ship's velocity, thereby ensuring that their round trip does not exceed 10% of their lifetime. Relativistic speed calculations help ensure the feasibility of interstellar trips, a concept that pushes the boundaries of current technology and inspires future advancements in space travel.