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Chapter 37: Problem 47

A space probe is sent to the vicinity of the star Capella, which is 42.2 light-years from the earth. (A light-year is the distance light travels in a year.) The probe travels with a speed of 0.9930c. An astronaut recruit on board is 19 years old when the probe leaves the earth. What is her biological age when the probe reaches Capella?

Short Answer

Expert verified
The astronaut's biological age upon reaching Capella is approximately 23.9 years.

Step by step solution

01

Identify the Relevant Concept

This exercise involves relativistic effects on time due to high-speed travel. Specifically, it requires the use of time dilation in Special Relativity to determine the biological age of the astronaut.
02

Formula for Time Dilation

Use the time dilation formula: \[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \( t' \) is the proper time experienced by the astronaut (biological time), \( t \) is the time experienced on Earth, \( v \) is the velocity of the probe, and \( c \) is the speed of light.
03

Calculate the Earth Time

First, calculate the time it takes for the probe to reach Capella from the perspective of Earth. Since Capella is 42.2 light-years away, and the probe speed is \( 0.9930c \), the time taken according to Earth is:\[ t = \frac{42.2}{0.9930} \approx 42.487 \text{ years} \]
04

Calculate the Proper Time (Astronaut's Time)

Substitute the values into the time dilation formula to find the astronaut's time:\[ t' = \frac{42.487}{\sqrt{1 - 0.9930^2}} \approx \frac{42.487}{\sqrt{1 - 0.9861}} \approx \frac{42.487}{0.1155} \approx 4.904 \text{ years} \]
05

Determine Astronaut's Biological Age

The recruit starts the journey at 19 years old. Therefore, her biological age when reaching Capella is:\[ 19 + 4.904 = 23.904 \] After rounding, it is approximately 23.9 years.

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Key Concepts

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

Time Dilation
Time dilation is a fascinating concept in the world of physics, particularly within the realm of special relativity. This idea was first introduced by Albert Einstein, who showed that time is not an absolute measure but can vary depending on the relative velocity of an observer.
Time dilation refers to the phenomenon where time, as measured by an observer in a different inertial frame, moves slower compared to a stationary observer. This effect is most noticeable at speeds approaching the speed of light.
  • For example, in the context of our space probe traveling at 0.9930 times the speed of light, the time experienced by the passenger (astronaut) on the probe differs from the time experienced on Earth.
  • The formula used to describe this relationship is: \[t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}\]where \( t' \) is the proper time experienced by the moving observer, \( t \) is the time in the stationary observer's frame (Earth), \( v \) is the velocity of the moving observer, and \( c \) is the speed of light.
  • This formula implies that as the velocity \( v \) becomes closer to \( c \), the denominator gets smaller, causing \( t' \) to decrease compared to \( t \).
Thus, for astronauts traveling at high speeds, aging occurs more slowly relative to those they left behind on Earth.
Relativistic Effects
Relativistic effects are crucial in understanding how objects behave at speeds that are significant compared to the speed of light. These effects become prominent in space travel scenarios, as in our example with the astronaut traveling to Capella.
Relativistic effects manifest mainly in three domains: length contraction, time dilation, and mass increase. However, for the purpose of our problem, the focus is on time dilation.
  • At nearly the speed of light, time dilation ensures that the astronaut feels less time passing on board the spaceship than the time calculated on Earth.
  • This difference can lead to various interesting and sometimes counterintuitive results, such as aging more slowly, as evidenced by the lower biological age of the astronaut by the journey's end.
  • Without accounting for these relativistic effects, calculations of time and distance in high-speed environments would be vastly inaccurate.
In situations like high-speed space travel, understanding relativistic effects is essential to accurately predict temporal experiences and changes.
Proper Time
Proper time is a fundamental concept in relativity that helps us understand how different observers in different frames measure the passage of time.
Proper time refers to the actual time interval measured by an observer as they travel along their worldline in spacetime. This is the time experienced by someone on a journey, such as our astronaut within the moving space probe traveling toward Capella.
  • In the context of the exercise, the proper time \( t' \) represents the amount of time the astronaut feels has passed according to her biological clock onboard the spaceship.
  • It is distinguished from the time experienced by an observer who is not moving with respect to the stars, termed as coordinate time \( t \). In this case, the time calculated on Earth.
  • Proper time is always less than or equal to the coordinate time for an observer in motion relative to a stationary observer.
Proper time allows us to make sense of the journey's duration experienced personally by the traveler, highlighting how speed can affect the perception of time itself.

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