Problem 15 - College Physics

Question

A A muon is created 55.0 \(\mathrm{km}\) above the surface of the earth (as measured in the earth's frame). The average life- time of a muon, measured in its own rest frame, is 2.20\(\mu \mathrm{s}\) , and the muon we are considering has this lifetime. In the frame of the muon, the earth is moving toward the muon with a speed of 0.9860\(c\) . (a) In the muon's frame, what is its initial height above the surface of the earth? (b) In the muon's frame, how much closer does the earth get during the lifetime of the muon? What fraction is this of the muon's original height, as measured in the muon's frame? (c) In the earth's frame, what is the lifetime of the muon? In the earth's frame, how far does the muon travel during its life- time? What fraction is this of the muon's original height in the earth's frame?

Step-by-Step Solution

Answer

(a) 9,068 m; (b) 651 m, 0.072; (c) 16.6 μs, 49100 m, 0.892.

Step 1: Understand the problem

We are given a muon created 55 km above the Earth. The muon's lifetime in its frame is 2.20 μs, and the muon perceives Earth moving towards it at 0.9860c. We need to find the initial height, distance traveled, and various lifetimes using both the muon's and Earth's frames.

Step 2: Convert given quantities and define variables

First, convert the height from km to meters: 55.0 km = 55,000 m. The speed of light is represented as \(c\) and equals \(3.00 \times 10^8 \, \text{m/s}\). Define the velocity \(v = 0.9860c\), and the muon's proper time (lifetime in its frame) \(\tau = 2.20 \, \mu\text{s} = 2.20 \times 10^{-6} \, \text{s}\).

Step 3: Calculate initial height in the muon's frame (Lorentz contraction)

Due to relativistic effects, the height as observed in the muon's frame is Lorentz contracted. Use the formula for length contraction: \(L = L_0 \sqrt{1 - \frac{v^2}{c^2}}\). Here, \(L_0 = 55,000 \, \text{m}\). Calculate the contracted height \(L\).

Step 4: Calculate how much closer Earth gets to the muon in its frame

In the muon's frame, the distance Earth gets closer is given by \(d = v \tau\). Compute \(d\) using the velocity \(v = 0.9860c\) and the muon's proper time \(\tau = 2.20 \times 10^{-6} \, \text{s}\).

Step 5: Find the fraction in the muon's original height (muon's frame)

The fraction can be found using \(\frac{d}{L}\), where \(d\) is the distance Earth moved closer, and \(L\) is the contracted height.

Step 6: Determine the lifetime of the muon in the Earth's frame (time dilation)

The time dilation formula is \(t = \tau / \sqrt{1 - \frac{v^2}{c^2}}\). Use this to find the lifetime \(t\) of the muon as measured in the Earth's frame.

Step 7: Calculate the distance the muon travels in the Earth's frame

In the Earth's frame, the muon travels a distance \(D = v \times t\), where \(t\) is the dilated time. Calculate \(D\) using the values found.

Step 8: Find the fraction of distance traveled to original height in Earth's frame

Compute the fraction \(\frac{D}{L_0}\), where \(L_0 = 55,000 \, \text{m}\) is the initial height in Earth’s frame.

Key Concepts

Lorentz Contraction

Lorentz contraction is a fascinating concept in the world of physics that deals with how lengths can appear shorter in a moving frame of reference. This contraction, or shortening of length, is a direct consequence of Einstein's theory of special relativity. Let's break it down simply: when an object moves at a significant fraction of the speed of light relative to an observer, its length along the direction of motion is measured to be shorter by that observer.

This phenomenon is described by the Lorentz contraction formula: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]where:

• \(L\) is the contracted length as measured by the observer,
• \(L_0\) is the proper length (the object's length in its own rest frame),
• \(v\) is the velocity of the object relative to the observer,
• \(c\) is the speed of light.

So, when a muon is created high above the Earth's surface, and it's moving toward Earth, we perceive its height above ground differently than if it were still. This relative motion makes the muon appear closer than it actually is in its own frame of reference. Understanding Lorentz contraction helps us predict and measure physical quantities accurately in high-speed environments.

Time Dilation

Time dilation is another key concept of special relativity, where time intervals are perceived to be different in different frames of reference. When an object moves close to the speed of light, time effectively slows down for it from the perspective of a stationary observer.

The formula that describes time dilation is:\[ t = \frac{\tau}{\sqrt{1 - \frac{v^2}{c^2}}} \]where:

• \(t\) is the time interval measured in the stationary observer's frame (dilated time),
• \(\tau\) is the proper time (the time interval measured in the moving object's frame),
• \(v\) is the velocity of the moving object,
• \(c\) is the speed of light.

In our muon exercise, the lifetime of the muon as it moves towards Earth is longer from the Earth's frame than in its own rest frame. This is because time dilation allows the muon to "live" longer as seen from Earth, meaning it can travel farther than if it were stationary relative to an observer on Earth. This highlights how motion can cause time to behave differently depending on your frame of reference.

Muon Lifetime

Muon lifetime is an interesting application of relativity principles, showcasing both Lorentz contraction and time dilation. In the exercise, muons are particles created high in the Earth's atmosphere and move very fast toward the surface. Their proper lifetime, the time they would "live" if they were at rest, is around 2.20 microseconds. However, due to their high velocities, they experience significant time dilation.

When considering the muon's frame, its lifetime is 2.20 microseconds, during which Earth moves towards it. Because of time dilation, this lifetime is longer when measured on Earth. These relativistic effects allow muons to reach the Earth's surface despite their short proper lifetime. This means that more muons reach the Earth's surface than expected when only classical (non-relativistic) physics is considered.

This intriguing aspect of relativity demonstrates how muons, which would otherwise decay long before hitting the ground, can actually be detected at sea level. This real-world scenario is a beautiful proof of Einstein's theories in action and provides a deeper understanding of how particles behave under relativistic conditions.

Frame of Reference

A frame of reference is a crucial part of understanding relativity and how events are perceived differently depending on where you stand, literally and figuratively. In simple terms, a frame of reference is the vantage point or context you use to observe phenomena. Each observer's frame can significantly affect measurements of speed, time, and distance.

For instance, in the exercise, a muon's frame is considered, where Earth is moving towards the muon. Conversely, in the Earth's frame, the muon is moving towards the Earth. Both of these perspectives are correct within their own contexts, but they yield different measurements due to relativistic effects.

This concept highlights how there is no single "correct" perspective in relativity. Instead, every frame of reference is valid, and the laws of physics work consistently within each one. This principle is foundational in relativistic physics, requiring us to account for differing observations and reconcile them using equations like Lorentz contraction and time dilation. Understanding frames of reference is essential in predicting and calculating outcomes in physics, particularly in scenarios involving high velocities close to the speed of light. It teaches us to appreciate the relative nature of observation and measurement in our universe.