Problem 52 - University Physics with Modern 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 \(\mathrm{c}\) . 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?

Step-by-Step Solution

Answer

The astronaut will be about 25 years old when the probe reaches Capella.

Step 1: Understand the Problem

The probe is traveling to Capella, which is 42.2 light-years away, at 99.10% the speed of light (denoted as 0.9910 \(c\)). You're asked to find out how many years will pass for an astronaut on the probe by the time it reaches its destination.

Step 2: Calculate Time According to Earth Observers

First, calculate the time it takes for the probe to reach Capella from the perspective of someone on Earth. Use the formula \( t = \frac{d}{v} \), where \( d \) is the distance (42.2 light-years) and \( v \) is the velocity (0.9910\( c \)).\[t_{\text{earth}} = \frac{42.2 \text{ light-years}}{0.9910c} = \frac{42.2}{0.9910} \approx 42.57 \text{ years}\]

Step 3: Time Dilation Formula

Since the astronaut is traveling at relativistic speeds, we need to account for time dilation. The time experienced by someone moving at velocity \( v \) is given by \( t' = t \sqrt{1 - \left( \frac{v}{c} \right)^2} \).\[t'_{\text{astronaut}} = 42.57 \sqrt{1 - (0.9910)^2}\]

Step 4: Approximate the Time Dilation Effect

Calculate the expression inside the square root:\[ 1 - (0.9910)^2 = 1 - 0.982081 = 0.017919 \]Plug this back into the formula:\[t'_{\text{astronaut}} = 42.57 \times \sqrt{0.017919} \approx 42.57 \times 0.1338 \]

Step 5: Calculate the Biological Time

Now find the product to get the dilated time experienced by the astronaut:\[ t'_{\text{astronaut}} \approx 5.694 \text{ years}\]

Step 6: Find the Astronaut's Biological Age

The astronaut was 19 years old when the probe began its journey. Add the time that passed relative to the astronaut to find her age when reaching Capella:\[ 19 + 5.694 \approx 24.694 \]So, her biological age will be approximately 25 years when rounded to the nearest whole number.

Key Concepts

Space Travel

Space travel in the context of our exercise involves a journey through space to a distant star, Capella, which is 42.2 light-years away. Understanding a light-year is crucial here. A light-year is the distance that light travels in one year, moving at about 299,792 kilometers per second. In this space travel scenario, we have a probe traveling to Capella. This journey is significant in illustrating the vast distances involved in space travel and how it can influence our understanding of time. To manage these large distances, the probe travels at relativistic speeds, near the speed of light. This allows us to explore interesting phenomena such as time dilation. Space travel at such high velocities is important not only for theoretical physics but also for understanding potential future human expeditions into deep space, where relativistic effects would become significant.

Speed of Light

The speed of light, often denoted as 'c', is a fundamental constant in physics, approximately equal to 299,792 kilometers per second. In the realm of physics and especially special relativity, the speed of light serves as an absolute speed limit. In our exercise, the probe travels at 0.9910 times the speed of light. This near-light speed is referred to as relativistic speed. As an object approaches this speed, relativistic effects like time dilation and length contraction become prominent.

• The speed of light is not dependent on the motion of the observer or the source of light, making it unique among speeds.
• Traveling at speeds close to 'c' results in time moving slower for the traveler compared to stationary observers.

This is central to understanding why the astronaut aboard the probe ages differently compared to observers on Earth. As speeds approach that of light, the laws of classical mechanics give way to those of relativistic physics, changing our perception of time and space.

Special Relativity

Special relativity, a theory developed by Albert Einstein, dramatically changed our understanding of time and space. One of the key postulates of this theory is that the laws of physics are invariant (the same) in all inertial frames of reference, and the speed of light is the same for all observers regardless of their relative motion. This has profound implications, one being time dilation.Time dilation comes into play when dealing with high-speed travel, like the probe journeying to Capella at 0.9910c. As the probe moves at such a significant fraction of the speed of light, time for the astronaut on board dilates (or stretches) compared to the time experienced by observers on Earth.

• According to special relativity, the moving astronaut experiences less passage of time.
• This is calculated using the Lorentz factor, derived from the equation \( t' = t \sqrt{1 - \left(\frac{v}{c}\right)^2} \).

Thus, despite the journey taking over 42 years from Earth's perspective, the astronaut ages only about 6 years, showcasing the relativistic effect of time dilation predicted by special relativity.