91Ó°ÊÓ

Space pilot Mavis zips past Stanley at a constant speed relative to him of 0.800\(c .\) Mavis and Stanley start timers at zero when the front of Mavis's ship is directly above Stanley. When Mavis reads 5.00 s on her timer, she turns on a bright light under the front of her spaceship. (a) Use the Lorentz coordinate transformation derived in Example 37.6 to calculate \(x\) and \(t\) as measured by Stanley for the event of turning on the light. (b) Use the time dilation formula, Eq. \((37.6),\) to calculate the time interval between the two events (the front of the spaceship passing overhead and turning on the light) as measured by Stanley. Compare to the value of \(t\) you calculated in part (a). (c) Multiply the time interval by Mavis's speed, both as measured by Stanley, to calculate the distance she has traveled as measured by him when the light turns on. Compare to the value of \(x\) you calculated in part (a).

Step by step solution

01

Understanding the Problem

We need to find the time and space coordinates of an event (turning on a light) from two different reference frames: Mavis's and Stanley's. Mavis moves at 0.800c relative to Stanley, and she activates the light when her timer reads 5.00 seconds.

02

Lorentz Transformation Setup

The Lorentz transformation equations are used to relate the space and time coordinates of events as observed in different frames. The equations are:\[ x' = \gamma (x - vt) \quad \text{and} \quad t' = \gamma \left(t - \frac{vx}{c^2}\right) \]where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \), \( x' \) and \( t' \) are coordinates in Stanley's frame, \( x \) and \( t \) are in Mavis's frame, and \( v = 0.800c \).

03

Calculate \( \gamma \)

First, calculate the Lorentz factor \( \gamma \) with Mavis's velocity \( v = 0.800c \):\[ \gamma = \frac{1}{\sqrt{1 - (0.800)^2}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} \approx 1.667 \]

04

Apply Lorentz Transformation to Find \(t\) and \(x\)

Using the initial condition \( x = 0 \) and \( t = 5 \) s in Mavis's frame, we apply the Lorentz transformation:1. \( t' = \gamma (t - \frac{vx}{c^2}) = 1.667 \times 5 = 8.335 \) s since \( x = 0 \) for the event.2. \( x' = \gamma (x - vt) = 1.667 \times (0 - 0.800c \times 5) = -6.67c \) seconds, which corresponds to the distance given the velocity.

05

Use Time Dilation to Find \( t' \)

Using time dilation given by the formula \( \Delta t' = \gamma \Delta t \), where \( \Delta t = 5 \) s in Mavis's frame:\[ \Delta t' = 1.667 \times 5 = 8.335 \text{ s} \]This matches the result for \( t' \) obtained from the Lorentz transformation.

06

Calculate Distance Traveled as Measured by Stanley

The distance covered by Mavis as measured by Stanley is given by \( x' = v \times t' = 0.800c \times 8.335 \) s = 6.67c seconds, consistent with part (a). This confirms the transformation.

07

Summarize and Verify Calculations

Both methods, Lorentz transformation and the direct calculation method (time dilation and speed calculation), should yield the same results. The calculations for \( t' = 8.335 \) s and \( x' = 6.67 \) confirm the analysis was done correctly.

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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 concept from Einstein's theory of special relativity. It describes how time can pass at different rates in different reference frames when they are moving relative to each other. For example, if you're in a fast-moving spaceship like Mavis in the original exercise, time for you will appear to pass more slowly than for someone stationary like Stanley on Earth. This means that 5 seconds for Mavis might equate to a longer duration, say 8.335 seconds for Stanley.

This phenomenon is quantified by the time dilation formula, \( \Delta t' = \gamma \Delta t \), where:

• \( \Delta t \) is the time interval measured in the moving frame (Mavis's frame).
• \( \Delta t' \) is the time interval in the stationary frame (Stanley's frame).
• \( \gamma \) is the Lorentz factor, calculated as \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \).

In this scenario, since Mavis moves at 0.800 times the speed of light, her experiences of time become stretched or dilated relative to Stanley's observations.

Reference Frames

Reference frames are essential to understanding how motion and events are measured differently from varied perspectives. A reference frame is essentially a point of view or a set of coordinates that helps define the position and time of events in space.

In physics, especially in situations involving high speeds close to that of light, the differences in observations from different reference frames become significant. In the earlier exercise, Mavis's and Stanley's frames are considered. Mavis's frame moves with her at 0.800 times the speed of light, while Stanley's frame is stationary relative to her motion.

Choosing a reference frame alters how we perceive movement, time intervals, and distances. For example:

• In Mavis's frame, time and space events occur at her observed rate.
• In Stanley's frame, these same events might happen at a slower rate or in different spatial positions due to relative motion.

Special Relativity

Special relativity is a groundbreaking theory proposed by Albert Einstein in 1905. It fundamentally changed our understanding of physics at high velocities. It is famous for challenging the classical mechanics assumptions and introducing new concepts like the constancy of the speed of light and the relativity of time.

Key principles include:

• The laws of physics are the same in all inertial reference frames, meaning that whether you are at rest or moving uniformly, the physical laws apply equally.
• The speed of light in a vacuum is always constant at approximately 299,792 kilometers per second, regardless of the observer's motion.

Special relativity brought about several results such as time dilation and length contraction. It laid the groundwork for modern physics and provided the framework for understanding phenomena on cosmic and quantum scales.

Speed of Light

The speed of light is a fundamental constant in physics, often denoted by \(c\) and is used as a pivotal benchmark for measuring high speeds in relativity theory. Light travels at about 299,792 kilometers per second in a vacuum. This speed remains constant across all reference frames according to the principles of special relativity.

In the exercise, Mavis's spaceship travels at 0.800c, meaning 80% of light's speed. This high velocity significantly affects time passage and space distances for events observed by both Mavis and Stanley. The constancy of light speed leads to various relativistic effects like the aforementioned time dilation and the invariance of physical laws.

Understanding the speed of light is crucial as it forms a core aspect of modern physics, not just in relativistic contexts but also in technologies and scientific calculations.