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Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Suppose that a spaceship was launched in the year 2120 on a round-trip journey of 100 light-years, traveling at \(99.99 \%\) of the speed of light. If one of the crew members was 30 years old when she left, about how old would you expect her to be on her return? (a) \(31 ;\) (b) \(130 ;\) (c) 29.

Short Answer

Expert verified
(a) 31; due to time dilation, she ages about 1 year during the trip.

Step by step solution

01

Understand the Problem

A spaceship travels 100 light-years at 99.99% of the speed of light. We need to determine the age of a crew member on her return, given she was 30 when she left.
02

Identify Relevant Concepts

The key concept to use here is 'time dilation' from the theory of relativity, which tells us that time for travelers moving close to the speed of light will elapse differently compared to stationary observers.
03

Calculate the Time as Per Earth Observers

Given the spaceship travels at 99.99% of the speed of light, the round trip would appear to take 100 years from Earth's perspective because it covers a distance of 100 light-years.
04

Apply Time Dilation Formula

Use the time dilation formula: \[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \(t'\) is the dilated time (time experienced by the crew), \(t\) is the time observed from Earth, \(v\) is the velocity of the spaceship, and \(c\) is the speed of light.Substitute \(v = 0.9999c\) to find:\[ t' = 100 \times \sqrt{1 - (0.9999)^2} \]
05

Solve for Dilated Time

Calculate \(t'\), which in this scenario is significantly smaller than Earth time. Upon solving for \(t'\), you find that the elapsed time for the crew is much less, approximately 1 year as they travel extremely close to light speed.
06

Determine Crew Member's Age on Return

The crew member's age is the sum of her starting age and the time elapsed for her during the travel. Therefore, she is 30 years old + 1 year = 31 years old when she returns.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Theory of Relativity
The theory of relativity, proposed by Albert Einstein, revolutionized our understanding of space and time. It comprises two theories: special relativity and general relativity. In the context of space travel, special relativity is most relevant. This theory suggests that the laws of physics are the same for all non-accelerating observers, and it introduces the concept that space and time are interconnected in a four-dimensional continuum known as spacetime.

One key aspect of special relativity is that the speed of light is constant in a vacuum for all observers, regardless of their motion relative to the light source. This leads to fascinating phenomena such as time dilation. As objects approach the speed of light, time for these objects appears to slow down relative to stationary observers. This is a fundamental consideration when discussing the aging of space travelers on high-speed journeys.
Speed of Light
The speed of light, denoted as \(c\), is approximately \(299,792,458\) meters per second in a vacuum. This speed is not just fast—it is a universal constant that serves as a cosmic speed limit. According to Einstein's theory of relativity, nothing can travel faster than the speed of light.

In the context of our spaceship journey, traveling at 99.99% the speed of light, the crew members experience significant relativistic effects. As they approach this speed, the effects predicted by special relativity, like time dilation, become apparent. The closer an object's speed to that of light, the more pronounced these effects are. Understanding the implications of reaching such high speeds is crucial when calculating phenomena like the elapsed time for travelers.
Round-Trip Journey
A round-trip journey in space involves traveling to a distant location and then returning to the starting point. In our exercise, the spaceship travels 100 light-years away and back, totaling 200 light-years of distance covered. In this context, light-years represent the distance that light can travel in one year, emphasizing the vastness of space.

From the perspective of observers on Earth, this journey seems to take a long time, because they measure it in years that light, the fastest thing in the universe, takes to travel. However, for the travelers on the spaceship moving at near-light speed, due to time dilation, the trip is considerably shorter from their own perspective. This difference in elapsed time for travelers and observers at rest stems from the relativistic effects described by the theory of relativity.
Elapsed Time for Travelers
Elapsed time for travelers is an intriguing concept arising from time dilation. In our scenario, although the journey takes 100 years from the perspective of Earth observers, the travelers themselves experience a much shorter duration.

To calculate this time, we used the time dilation formula: \[t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}\] Here, \(t'\) is the time experienced on the spaceship (the travelers), \(t\) is the time as seen by stationary observers (Earth), \(v\) is the ship's velocity, and \(c\) is the speed of light.

With a velocity of 99.99% of the speed of light, the calculation shows that the time elapsed for the crew is approximately just 1 year. This outcome illustrates the powerful effects of traveling at relativistic speeds, where a century on Earth can pass while only a single year is counted by those on board the spaceship. Consequently, the crew member who was 30 at departure would only be 31 upon return, highlighting time's pliability at high velocities.

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