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Mastering Length Contraction & Relativistic Physics
Dive into the fascinating world of special relativity! Understand length contraction, relativistic momentum, and the famous E=mc² equation. Challenge your perceptions and grasp these revolutionary physics concepts.

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Now Playing:Length contraction and relativistic momentum mass and energy1 – Example 0a
Intros
  1. Length Contraction & Relativistic Momentum, Mass and Energy
  2. Length Contraction & Relativistic Momentum, Mass and Energy
    Introduction to length contraction
  3. Length Contraction & Relativistic Momentum, Mass and Energy
    Relativistic Momentum, Mass, and Energy
Examples
  1. A star is 62 light-years away in the Earth frame of reference. How fast would you have to travel so that to you the distance is only 15 light-years?
    Length contraction & relativistic momentum, mass and energy
    Jump to:NotesConceptFAQsPrerequisites
    Notes

    In this lesson, we will learn:

    • How to find the length contraction?
    • How to calculate the relativistic momentum?
    • The relation between total energy, kinetic energy and rest energy
    • Relativistic Addition of Velocities

    Notes:

    In addition to the time interval difference in diverse frames of references, lengths and distances also vary from one frame to another.

    Let’s consider a spaceship travelling at very high speed from Earth to another planet, and two different frame of references to calculate the length contraction.

    1. Observer on earth

      2. Length Contraction and Relativistic Momentum, Mass and Energy

      L0 L_{0}: Distance between plants as measured by the observer (proper length)
      △t\triangle t : Time required for the trip measured form Earth

      △t=L0v\triangle t = \frac{L_{0}}{v}

    2. Observer in the spaceship

      3. Length Contraction and Relativistic Momentum, Mass and Energy

    The time interval for the observer in the spacecraft is shorter, since the spacecraft is moving. So the time for the trip according to the time dilation equation would be;

    △t0=△t(1−v2/c2\triangle t_{0} = \triangle t \sqrt{(1 - v^{2} / c^{2} }

    LL: the distance between the plants as viewed by the spacecraft observer.

    L=v△t0=v△t(1−v2/c2)=L0(1−v2/c2)L = v\triangle t_{0} = v \triangle t\sqrt{(1 - v^{2} / c^{2})} = L_{0}\sqrt {(1 - v^{2} / c^{2})}

    Length Contraction and Relativistic Momentum, Mass and Energy


    γ=\gamma = 1(1−v2/c2)\large\frac{1}{\sqrt{(1 - v^{2} / c^{2}) }} ⇒L=L0γ \quad \Rightarrow \quad L = \frac{L_{0}} {\gamma}

    therefore; the length of the object is measured to be shorter when it is moving relative to the observer than when it is at rest. (length Contraction)

    Note: the length contraction occurs only along the direction of the motion. If the object is moving along xx-axis, the length is shortened but the height stays the same.

    For example, if there is a rectangular painting (1.00m by 1.50m) hung on the wall of a spaceship which is moving at a speed of 0.8c, contraction occurs along the length only.

    Length Contraction and Relativistic Momentum, Mass and Energy


    L=L0(1−v2/c2)= L = L_{0}\sqrt {(1 - v^{2} / c^{2}) } = (1.50mm) (1−(0.8c)2/c2)= \sqrt{(1 - (0.8c)^{2} / c^{2})} = 0.9mm


    Length Contraction and Relativistic Momentum, Mass and Energy


    Relativistic Momentum

    In non-relative situation momentum is equal to p=m0vp = m_{0}v (classical momentum),
    In relativistic situation the rest mass (m0m_{0}) increases with the speed and is called relativistic mass (mrelm_{rel})

    mrel= m_{rel} = m0(1−v2/c2) \large \frac{m_{0}} {\sqrt{(1 - v^{2} / c^{2})} } =γ m0 = \gamma \,m_{0}

    therefore, the relativistic momentum can be calculated using the following equation

    p=mrelv=γ m0v=p = m_{rel} v = \gamma \, m_{0} v = m0v(1−v2/c2)\large \frac{m_{0} v} {\sqrt{(1 - v^{2} / c{2})} }

    Mass and Energy

    The total amount of energy possessed in this case is the total kinetic energy and the rest energy, Etotal=Ek+E0E_{total} = E_{k} + E_{0}

    The rest energy by Einstein is defined as E0=m0c2E_{0} = m_{0} c^{2}

    The total energy is also calculated by Einstein’s equation but the mass should be

    the relativistic mass Etotal=mrel c2=E_{total} =m_{rel} \, c^{2} = m0 c2(1−v2/c2) \large \frac{m_{0} \,c^{2}} {\sqrt{(1 - v^{2} /c^{2}) }}

    Ek=Etotal − E0= E_{k} = E_{total} \,- \, E_{0} = m0 c2(1−v2/c2)\large \frac{m_{0} \, c^{2}} {\sqrt{(1 - v^{2} / c^{2}) }} − m0 c2=m0c2 - \,m_{0} \,c^{2} = m_{0} c^{2}(1(1−v2/c2)−1)\large (\frac{ 1 } { \sqrt{(1 - v^{2} / c^{2})} } - 1)

    Ek=m0c2E_{k} = m_{0}c^{2} (1(1−v2/c2)−1)\large (\frac{1}{\sqrt{(1 - v^{2} / c^{2}})} - 1)


    Relativistic Addition of Velocities
    Relative Velocities do not add simply, as in classical mechanics (vv << cc)
    Length Contraction and Relativistic Momentum, Mass and Energy


    u=v + u′1 + v u′/c2 u = \frac{v \, + \, u^{\prime} }{1 \, + \,v \,u^{\prime} / c^{2}} \qquad (u′u^{\prime} and vv are in the same direction)
    Concept

    Introduction to Special Relativity Concepts

    Welcome to our exploration of special relativity! In this section, we'll dive into fascinating concepts like length contraction, relativistic momentum, mass, and energy. These ideas revolutionized physics and our understanding of the universe. The introduction video you'll watch is crucial for grasping these complex topics. Length contraction describes how objects appear shorter when moving at high speeds. Relativistic momentum explains how an object's momentum increases as it approaches the speed of light. Mass and energy are interconnected through Einstein's famous equation E=mc². This relationship shows that mass can be converted to energy and vice versa. As we progress, you'll see how these concepts challenge our everyday perceptions of space and time. Remember, while these ideas might seem counterintuitive at first, they're supported by extensive experimental evidence. Let's embark on this exciting journey to unravel the mysteries of special relativity together!

    FAQs

    1. What is length contraction and how does it work?

      Length contraction is a phenomenon in special relativity where objects appear shorter when moving at high speeds relative to an observer. It's described by the equation L = L (1 - v²/c²), where L is the observed length, L is the proper length, v is the relative velocity, and c is the speed of light. As an object's speed approaches the speed of light, its observed length decreases along the direction of motion.

    2. How does relativistic momentum differ from classical momentum?

      Relativistic momentum (p = γmv) differs from classical momentum (p = mv) by including the Lorentz factor γ = 1 / (1 - v²/c²). This factor accounts for the increase in an object's apparent mass as its velocity approaches the speed of light. As a result, relativistic momentum increases more rapidly with velocity than classical momentum, approaching infinity as the speed nears that of light.

    3. What does E=mc² mean and why is it important?

      E=mc² is Einstein's famous equation expressing mass-energy equivalence. It states that energy (E) is equal to mass (m) multiplied by the speed of light (c) squared. This equation reveals that mass and energy are interchangeable forms of the same thing, with even small amounts of mass equivalent to enormous amounts of energy. It's crucial in understanding nuclear reactions, particle physics, and the fundamental nature of the universe.

    4. How does relativistic velocity addition differ from classical velocity addition?

      Relativistic velocity addition uses the formula v = (u + v') / (1 + uv'/c²), where v is the relative velocity between two objects, u and v' are the individual velocities, and c is the speed of light. Unlike classical addition, this ensures that the combined velocity never exceeds the speed of light. For everyday speeds, it approximates classical addition, but at high velocities, it prevents violations of the universal speed limit.

    5. What are some practical applications of special relativity in everyday life?

      Special relativity has several practical applications, including: 1) GPS systems, which require relativistic corrections for accurate positioning. 2) Particle accelerators, where relativistic effects are crucial for proper operation. 3) Atomic clocks, which account for relativistic time dilation for precise timekeeping. 4) Nuclear energy production, based on the principle of mass-energy equivalence. These applications demonstrate how this seemingly abstract theory impacts technology and our daily lives.

    Prerequisites

    Understanding the foundations of physics is crucial when delving into advanced concepts like length contraction and relativistic momentum, mass, and energy. One of the most fundamental prerequisites for grasping these complex ideas is a solid understanding of relative velocity. This concept serves as a cornerstone for comprehending the intricacies of special relativity and its implications on our perception of space, time, and motion.

    Relative velocity is essential because it introduces the idea that motion is not absolute but depends on the observer's frame of reference. This concept directly relates to length contraction and relativistic effects, as these phenomena occur when objects move at speeds approaching the speed of light relative to an observer. By mastering relative velocity, students can more easily grasp how the perception of an object's length or its momentum can change based on the relative motion between the object and the observer.

    Moreover, a strong foundation in relative velocity helps in understanding the principle of relativity, which states that the laws of physics are the same in all inertial reference frames. This principle is fundamental to the development of special relativity, which in turn leads to the concepts of length contraction and relativistic momentum, mass, and energy.

    When studying length contraction, students will find that their knowledge of relative velocity provides a crucial context for understanding why objects appear to shrink in the direction of motion when observed from a different reference frame. Similarly, in exploring relativistic momentum and energy, the concept of relative velocity helps explain why classical Newtonian mechanics breaks down at high speeds and why we need to consider relativistic effects.

    The relationship between energy and mass, famously expressed in Einstein's equation E = mc², also builds upon the foundation of relative velocity. Understanding how velocity affects the kinetic energy of an object at relativistic speeds becomes much more intuitive when students have a solid grasp of relative motion.

    In conclusion, mastering relative velocity is not just a prerequisite but a gateway to comprehending the more advanced topics of length contraction and relativistic effects. It provides the necessary conceptual framework for students to navigate the counterintuitive world of special relativity and appreciate the profound implications it has on our understanding of the universe. By investing time in thoroughly understanding this fundamental concept, students will find themselves better equipped to tackle the challenges and revelations that come with studying relativistic physics.