Problem 13 - Fundamentals of Physics

Question

A space traveler takes off from Earth and moves at speed \(0.9900 c\) toward the star Vega, which is 26.00 ly distant. How much time will have elapsed by Earth clocks (a) when the traveler reaches Vega and (b) when Earth observers receive word from the traveler that she has arrived? (c) How much older will Earth observers calculate the traveler to be (measured from her frame) when she reaches Vega than she was when she started the trip?

Step-by-Step Solution

Answer

(a) 26.26 years, (b) 52.26 years, (c) 3.705 years.

Step 1: Calculate Time Elapsed on Earth's Clock When Traveler Reaches Vega

The distance to Vega is given as 26.00 light-years and the speed of the traveler is \(0.9900c\). Using the formula for time, \( t = \frac{d}{v} \), where \( d \) is the distance, and \( v \) is the speed of the traveler, we find the time elapsed on Earth clocks. \[ t = \frac{26.00 \, \text{ly}}{0.9900c} = \frac{26.00}{0.9900} \, \text{years} \approx 26.26 \, \text{years} \] Thus, when the traveler reaches Vega, 26.26 years will have elapsed on Earth clocks.

Step 2: Calculate Time Elapsed When Earth Receives Message from Traveler

Once the traveler reaches Vega, the message sent back to Earth travels at the speed of light. Since Vega is 26.00 light-years away, it takes 26.00 years for the message to reach Earth. Thus, the total time elapsed by the time Earth observers receive the message is the sum of the time taken for the traveler to reach Vega and the time for the message to return:\[ 26.26 \, \text{years} + 26.00 \, \text{years} = 52.26 \, \text{years} \] Hence, Earth clocks will show 52.26 years elapsed when they receive the message.

Step 3: Calculate Traveler's Aging from Earth's Perspective

From the traveler's frame, the time experienced is different due to time dilation. Using the time dilation formula \( t' = \frac{t}{\gamma} \), where \( \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \), we calculate the time as perceived by the traveler.For \( v = 0.9900c \), we have:\[ \gamma = \frac{1}{\sqrt{1-(0.9900)^2}} \approx 7.088 \]So, the traveler's recorded time is:\[ t' = \frac{26.26 \, \text{years}}{7.088} \approx 3.705 \, \text{years} \]The traveler ages approximately 3.705 years from their own perspective when reaching Vega.

Key Concepts

Time Dilation

Time dilation is a fascinating phenomenon predicted by Einstein's theory of relativity. It occurs when an object approaches the speed of light. In time dilation, time is experienced differently by observers in different frames of motion. For instance, if a space traveler moves near the speed of light, they will experience time passing slower compared to those who remain stationary on Earth.

• When calculating time dilation, we use the Lorentz factor, often denoted as \( \gamma \).
• Time in the traveler's frame, \( t' \), relates to Earth's time, \( t \), via the equation \( t' = \frac{t}{\gamma} \).

This concept means the traveler to Vega perceives time passing rather quickly, aging slower by their own perspective.

Light-Year

A light-year is an astronomical unit of distance rather than time. It defines how far light travels in one year. Since light travels at an impressive speed of \(2.998 \times 10^8\) m/s, this means a light-year equals about 9.46 trillion kilometers.

• In the context of space travel, light-years allow us to easily convey vast cosmic distances.
• For our space traveler heading towards Vega, the destination is noted to be 26.00 light-years away.

Thinking in terms of light-years helps in understanding the truly vast distances between stars and galaxies in space. It encapsulates just how far away destinations like Vega truly are.

Speed of Light

The speed of light, denoted by \( c \), plays a crucial role in the fabric of space-time and relativity. In a vacuum, light speed is always \(2.998 \times 10^8\) meters per second (m/s). This speed is a cosmic speed limit; no object with mass can reach or exceed it.

• The speed of light paves the way for understanding the universe's fundamental limits.
• In our scenario, the traveler's speed is given as \(0.9900c\), meaning 99% of the speed of light.

Traveling close to this speed creates noticeable effects, such as the time dilation experienced by our traveler on their journey to Vega.

Lorentz Factor

The Lorentz factor, \( \gamma \), is an essential tool in special relativity. It helps us calculate phenomena like time dilation and length contraction when an object approaches relativistic speeds. The factor is calculated using the equation \( \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \), where \( v \) is the object's velocity, and \( c \) is the speed of light.

• The Lorentz factor increases significantly as a body's velocity approaches the speed of light.
• For our traveler's journey at 0.9900c, we find \( \gamma \approx 7.088 \).

This factor shows us how much time dilates and distances contract at high speeds, changing how the universe is perceived by differently moving observers.