Question

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 the time, the normal human lifetime is 400.00 years.) What is the minimum speed with which the ship carrying the expedition must travel?

Short answer

Maximum allowed time = \(0.1 \times 400\) years = 40 years The maximum allowed time for the round-trip expedition is 40 years. #tag_title#Step 2: Calculate the total distance to be traveled.#tag_content# Since it's a round-trip, the total distance to be traveled is twice the given distance. Total distance = 2 × 2000 light-years = 4000 light-years #tag_title#Step 3: Apply the Time formula to find the minimum speed.#tag_content# We can now use the Time formula, \(time = \frac{distance}{speed}\), to find the minimum speed necessary to complete the round-trip within the given time. Rearranging the formula to find the speed: Speed = \(\frac{distance}{time}\) Minimum speed = \(\frac{4000\, \text{light-years}}{40\, \text{years}}\) Calculating the minimum speed: Minimum speed = 100 light-years per year The minimum speed with which the ship should travel for the round-trip to be completed within 10% of a human lifetime is 100 light-years per year.

Step-by-step solution

Calculate the maximum allowed time for the round-trip expedition.

First, we need to find the maximum time allowed for the round-trip expedition, which is 10% of the normal human lifetime (400 years). Maximum allowed time = \(10.000\%\) of 400 years = \(\frac{10}{100} \times 400\) years Calculating the maximum allowed time:

Key concepts

Special Relativity

Understanding special relativity is key to grasping concepts relevant to interstellar travel. Developed by Albert Einstein in 1905, special relativity revolutionized the way we think about space, time, and how they're related. One of its central tenets is that the laws of physics are the same for all non-accelerating observers, and, notably, the speed of light in a vacuum is the same for all observers, no matter their motion relative to the light source.

This has profound implications for space travel. As speeds increase towards the speed of light, time, lengths, and mass cannot be viewed as absolute. To an observer at rest, a clock on a spaceship traveling near the speed of light would appear to tick more slowly, and the spaceship itself would appear to contract in length. These effects are mathematically described by Lorentz transformations, which showcase how time and space measurements change as a function of the relative speed between the observer and an object in motion. It's this framework that leads us into the phenomenon known as time dilation.

Time Dilation

Time dilation is a fascinating consequence of special relativity, directly impacting the thought exercise of interstellar travel. It refers to the difference in the elapsed time as measured by two clocks due to a relative velocity between them. Simply put, if you're traveling at a significant fraction of the speed of light, time will pass more slowly for you than for someone who remains stationary.

In the context of our interstellar travel exercise, the travelers on the spaceship bound for a star 2000.0 light-years away would experience time at a slower rate than people back on Earth. Therefore, even if centuries pass on Earth, the astronauts could experience time in such a compressed fashion that the round-trip could feel much shorter. The formula that illustrates this effect is known as the time dilation formula: \(\Delta t' = \Delta t / \sqrt{1 - v^2/c^2}\), where \(\Delta t'\) is the time interval measured by the moving observer, \(\Delta t\) is the time interval measured by the stationary observer, \(v\) is the relative velocity, and \(c\) is the speed of light. This relationship shows us that as the speed \(v\) increases, the ratio within the square root decreases, leading to a greater difference in time experienced by the stationary and moving observers, respectively.

Light-Year Measurement

The term 'light-year' is essential for understanding distances in space, and it is often a source of confusion because it refers to distance, not time. A light-year is the distance that light travels in a vacuum in one year. Since nothing can travel faster than light—about 299,792 kilometers per second—a light-year becomes a substantial measure. Specifically, one light-year equals nearly 9.461 trillion kilometers or about 5.879 trillion miles.

For our space expedition, the star they intend to visit is 2000.0 light-years away. This means that light from this star takes 2000.0 years to reach Earth. When considering this vast distance, the spacecraft's speed becomes crucial, as the travelers will need to move at a significant fraction of the speed of light to ensure the round-trip fits within the confines of human lifetimes, based on the parameters of their mission. To determine the necessary speed, we calculate not only with time dilation in mind but also considering the entirety of the distance to be covered—the substantial 2000.0 light years there and back again.