Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 38

Describe the shape of the world line on a spacetime diagram of (a) an object that remains at rest at a specific position along the \(x\) -axis; (b) an object that moves at constant velocity \(u\) in the \(x\) -direction; \((c)\) an object that begins at rest and accelerates at a constant rate of in the positive \(x\) -direction.

Short Answer

Expert verified
For the three given cases, the shape of the world lines on a spacetime diagram are as follows: (a) For an object at rest, the world line is a vertical line parallel to the time axis; (b) For an object moving at a constant velocity, the world line is a straight line with constant slope, inclined with respect to the time axis; (c) For an object initially at rest but accelerating in the positive x-direction, the world line is a parabola, with its curvature indicating acceleration.

Step by step solution

01

Case (a): Object at rest at a specific position along the x-axis

Since the object is at rest, its position remains constant along the x-axis. That means that for any given time, the object's position doesn't change. Thus, the world line for the object at rest will be a vertical line on the spacetime diagram, parallel to the time (y) axis.
02

Case (b): Object moving at a constant velocity u in the x-direction

An object moving at a constant velocity has a linear displacement with respect to time. Mathematically, the position of an object can be represented as \(x(t) = x_0 + ut\), where \(x_0\) is the initial position, \(u\) is the constant velocity, and \(t\) represents time. Since the position changes linearly with time, the world line for an object moving at a constant velocity will be a straight line with a constant slope, inclined with respect to the time (y) axis.
03

Case (c): Object begins at rest and accelerates at a constant rate in the positive x-direction

In this case, the object starts at rest and accelerates at a constant rate. This means that its position will change with respect to time in a non-linear manner. The position of an object under the influence of constant acceleration can be represented as \(x(t) = x_0 + \frac{1}{2}at^2\), where \(x_0\) is the initial position, \(a\) is the constant acceleration, and \(t\) represents time. In this case, the world line will be a curve (specifically, a parabola) on the spacetime diagram, with the curvature indicating acceleration.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

World Line
Imagine a graph that illustrates an object's journey through the universe, where time ticks upward and position extends horizontally. This is the essence of a spacetime diagram. A world line on such a diagram is like a storyline for an object, tracing its history from past to future.

A world line isn't just a simple plot of position versus time; it is the spine of a tale in the fabric of spacetime. For an object at rest, its tale is steady and unvarying – a vertical line standing tall and unwavering on the graph. This vertical line indicates that as time flows on, the object holds its ground, unchanging in location.
Constant Velocity
When an object cruises through space without speeding up or slowing down, its motion is captured by an unwavering slant on the spacetime diagram – the hallmark of a constant velocity. This linear world line is akin to the object leaving a breadcrumb trail at evenly spaced intervals, indicating a uniform pace through its journey.

The steadiness of its path tells us that its motion, unswayed by accelerations or decelerations, respects the symmetry of space and time. This kind of world line is a straight, angled path, distinct from the stationary vertical, revealing a consistent progression through both space and time.
Constant Acceleration
Enter the realm of acceleration, and our straight and narrow path begins to bend. Constant acceleration draws a curve on our spacetime diagram, a parabolic arc that speaks of a changing pace. This object's world line tells a more dynamic story; it starts at rest but then, as each moment passes, it gains speed.

This world line swells with the curvature of increasing velocity. Visualize a ball rolling down a gentle slope, gathering momentum – that's the very essence of constant acceleration captured on the graph, a testament to the object's growing hastiness.
Kinematics
The branch of physics that deals with motion, kinematics, is the storybook of how objects move through space and time, devoid of the causes behind the motion. It's akin to watching the ballet of celestial bodies without questioning the orchestra of forces conducting their dance.

Through equations of motion, kinematics lays out the rules of this ballet. Whether an object is at peace, slides through space with unfaltering speed, or hastens its stride, kinematics translates these tales into mathematical language, enabling us to predict where and when the next scene of the motion-play will unfold.
Special Relativity
Albert Einstein's theory of special relativity brought a seismic shift to our understanding of space and time. Among its many profound revelations, it shows us that the spacetime diagram is more than a mere plot; it’s a canvas where the speed of light is the ultimate artist, drawing the boundary of cosmic possibility.

Special relativity intertwines space and time, revealing that the faster an object moves, the more its experience of time alters. This theory adds depth to our spacetime diagrams and enriches the narrative of world lines, ensuring that the laws of physics hold true from the perspective of observers, no matter the speed at which they traverse the cosmos.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A neutral \(\pi\) -meson is a particle that can be created by accelerator beams. If one such particle lives \(1.40 \times 10^{-16} \mathrm{s} \quad\) as measured in the laboratory, and \(0.840 \times 10^{-16} \mathrm{s}\) when at rest relative to an observer, what is its velocity relative to the laboratory?

A man is running on a straight road that makes \(30^{\circ}\) with the train track. The man is running in the direction on the road that is away from the track at a speed of \(12 \mathrm{m} / \mathrm{s}\). The train is moving with a speed of \(30 \mathrm{m} / \mathrm{s}\) with respect to the track. What is the speed of the man with respect to a passenger sitting at rest in the train?

What is the velocity of an electron that has a momentum of \(3.04 \times 10^{-21} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) ? Note that you must calculate the velocity to at least four digits to see the difference from \(c\)

Find the velocity of a proton that has a momentum of \(4.48 \times 10^{-19} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\)

The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Kinematics Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Turning Points in Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Force

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free