Problem 38 - College Physics

Question

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 0.9910\(c\) relative to the earth. 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, as measured by (a) the astronaut and (b) someone on earth?

Step-by-Step Solution

Answer

She is 25.032 years old by her measure and 61.55 years old according to Earth observers.

Step 1: Calculate the time taken according to Earth observers

We first need to determine how long it takes for the probe to travel to Capella as seen by someone on Earth. Given that the distance to Capella is 42.2 light years and the velocity of the probe is 0.9910c, we use the formula \( t = \frac{d}{v} \). Thus, \( t = \frac{42.2}{0.9910} \approx 42.55 \text{ years} \).

Step 2: Calculate the time taken according to the astronaut onboard

Next, we calculate the time experienced by the astronaut using the time dilation formula from the theory of relativity: \( t' = t \sqrt{1 - \frac{v^2}{c^2}} \). Substituting the values, we have \( t' = 42.55 \sqrt{1 - (0.9910)^2} \approx 6.032 \text{ years} \).

Step 3: Determine the astronaut's age on Earth

The astronaut was 19 years old when the journey started. After the journey, according to Earth time, she would be \( 19 + 42.55 = 61.55 \text{ years old} \).

Step 4: Determine the astronaut's biological age

According to time experienced on the probe, the astronaut's age at arrival is \( 19 + 6.032 = 25.032 \text{ years old} \).

Key Concepts

Time Dilation

Time dilation is a fascinating concept from Einstein's theory of special relativity. Imagine you are traveling on a spaceship at a high speed. For an observer on Earth, time on the spaceship seems to move slower compared to time on Earth. This is what we refer to as time dilation. The faster you travel, the more pronounced this effect becomes.
Time dilation occurs because the speed of light is a constant in all frames of reference. According to the theory of relativity, the laws of physics are the same for everyone, regardless of how fast they are moving relative to each other. That's why, for the astronaut traveling at 0.9910 times the speed of light to Capella, time passes more slowly compared to an observer on Earth.

• The equation for time dilation is: \( t' = t \sqrt{1 - \frac{v^2}{c^2}} \), where \( t \) is the time measured by someone on Earth, \( v \) is the velocity of the moving object, and \( c \) is the speed of light.
• This effect is why the astronaut only ages about 6.032 years, while over 42 years pass on Earth during the journey to the star Capella.

Light Year

A light year is not a measurement of time, rather it is a measurement of distance. When we say something is a light year away, we mean that light takes one whole year to travel that distance.
Light travels at an incredibly fast speed of approximately 299,792,458 meters per second, or about 186,282 miles per second. Over a year, this adds up to an incredible distance. To put it into perspective, the distance between the Earth and the Sun is about 93 million miles, and light covers this distance in just over 8 minutes.

• A light year is equivalent to about 5.88 trillion miles or 9.46 trillion kilometers.
• This vast measure of distance underscores the enormous scales we're dealing with when discussing interstellar travel.
• When the exercise states Capella is 42.2 light years away, it implies the distance light travels in 42.2 years—truly, a cosmic journey.

Special Relativity

Special relativity is one of the ground-breaking theories introduced by Albert Einstein in 1905. It revolutionized how we understand motion, space, and time, particularly at high speeds.
The theory fundamentally changed how we think about space and time, merging them into a single entity known as space-time. One of its crucial postulates is that the speed of light in a vacuum is constant and is independent of the observer's motion and the motion of the light source.

• Special relativity introduces concepts like time dilation and length contraction, which become significant at speeds approaching the speed of light.
• It sets the limits on how fast anything in our universe can travel; nothing can exceed the speed of light, which is the ultimate speed limit.
• For space travel, this means that while we can imagine traveling vast distances to stars like Capella, the relative nature of time means we must consider the effect of high-speed travel on time and aging.

Space Travel

Space travel fundamentally transforms our conception of distance and time, especially when considering the relativistic effects at near-light speeds. Traveling to distant stars, such as Capella, requires technology and understanding that go beyond current capabilities, but it intrigued physicists and science fiction alike.
To explore the cosmos as depicted in this exercise, the theoretical boundaries outlined by special relativity become practical challenges. For astronauts, even though years might pass on Earth, their experience of time is significantly less due to time dilation. This would make planning and executing long-duration space missions a complex task.

• Space travel involves not just overcoming vast distances but also preparing for the physiological effects on humans, like the significant time differences experienced by travelers versus stationary observers on Earth.
• Developing propulsion technologies and life-support systems capable of sustaining human life over such long journeys is paramount.
• Despite such challenges, space travel holds tremendous potential for discovery and understanding of our universe.