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Length contraction & relativistic momentum, mass and energy

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Intros
Lessons
  1. Length Contraction & Relativistic Momentum, Mass and Energy
  2. Introduction to length contraction
  3. Relativistic Momentum, Mass, and Energy
  4. Relativistic Addition of Velocities
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Examples
Lessons
  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?
    1. Suppose you decided to travel to a star 60 light-years away at a speed that tells you the distance is only 15 light-years. How many years would it take to make the trip?
      1. At what speed will the length of a 1.00m stick look 20.0% shorter?
        1. The length of a ruler placed in spaceship in 32cm. The spaceship moves with a speed of 0.080c. what would be the percent decrease in the length of the ruler?
          1. What is the relativistic momentum of an antiproton moving at a speed of 0.68c and mass of 1.674 x 10-27 kg?
            1. What is the percentage change in the momentum of an electron that accelerates from 0.25c to 0.80c? (mass of electron = 9.12 x 10-31kg)
              1. Calculate the rest energy of an electron in joules and in Mev
                (1Mev= 1.60 x 10-13J)
                1. What would be the momentum and kinetic energy of a proton travelling with a speed of 1.80 x 108m/s. (Proton rest mass = 1.67 x 10-27kg)
                  1. Calculate the kinetic energy and momentum of a proton travelling 6.24 x 107m/s. What would be the percentage error if the value was calculated using classical formula?
                    1. A person on a spaceship traveling at 0.60c (with respect to the Erath) observers a comet come from behind and pass her at a speed of 0.60c. how fast is the comet moving with respect to the Earth?
                      1. A spaceship leaves the Earth traveling at 0.61c. A second spaceship leaves the first at a speed of 0.77c with respect to the first one. Calculate the speed of the second spaceship with respect to the earth if it is fired
                        1. In the same direction the first spaceship is already moving
                        2. Directly backwards towards the Earth
                        Topic Notes
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                        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!

                        Length Contraction

                        Length contraction is a fascinating phenomenon in special relativity that challenges our everyday understanding of space and time. This concept, first introduced by Albert Einstein, describes how objects appear shorter when they move at high speeds relative to an observer. To fully grasp this idea, let's dive into its derivation, significance, and real-world implications.

                        The derivation of the length contraction equation stems from Einstein's special theory of relativity. It begins with two fundamental postulates: the principle of relativity and the constancy of the speed of light. Using these principles, we can derive the Lorentz transformation equations, which relate space and time coordinates between different inertial frames of reference.

                        The length contraction equation is expressed as:

                        L = L (1 - v²/c²)

                        Where L is the observed length, L is the proper length (length in the object's rest frame), v is the relative velocity between the object and observer, and c is the speed of light. This equation shows that as an object's speed approaches the speed of light, its observed length decreases.

                        The significance of length contraction in special relativity cannot be overstated. It fundamentally alters our perception of space and time, demonstrating that these concepts are not absolute but relative to the observer's frame of reference. This revelation has profound implications for our understanding of the universe and has led to numerous scientific advancements.

                        To illustrate how length contraction affects objects moving at high speeds, let's consider the classic spaceship example. Imagine a spaceship traveling at 80% the speed of light past Earth. To an observer on Earth, the spaceship would appear shorter along its direction of motion compared to its length when at rest. This contraction is not an optical illusion but a real effect of the relative motion between the spaceship and Earth.

                        Another intriguing scenario to consider is the painting example. Picture a 10-meter-long painting hanging in a gallery. Now, imagine this painting could somehow move at 99% the speed of light relative to you. As you observe the painting zipping by, it would appear to be only about 1.4 meters long in the direction of motion. However, it's crucial to note that from the painting's perspective (if it could have one), it would still measure 10 meters long, and it would be the gallery that appears contracted.

                        It's important to understand that length contraction only occurs along the direction of motion. Objects do not contract in dimensions perpendicular to their movement. This means a spherical object would appear as an ellipsoid when moving at high speeds, compressed along its direction of travel.

                        While these effects may seem counterintuitive, they have been experimentally verified. For instance, the behavior of particles in particle accelerators aligns perfectly with the predictions of special relativity, including length contraction.

                        In our everyday lives, we don't notice length contraction because the speeds we typically encounter are far too low compared to the speed of light. The effects become noticeable only at speeds approaching a significant fraction of light speed. This is why length contraction is primarily relevant in the realm of particle physics and theoretical astrophysics.

                        Understanding length contraction is crucial for various scientific and technological applications. For example, it plays a role in GPS satellite calculations, ensuring accurate positioning. It's also essential in the design and operation of particle accelerators, where scientists must account for relativistic effects to accurately predict and interpret experimental results.

                        As we continue to push the boundaries of science and technology, concepts like length contraction become increasingly relevant. From exploring the nature of black holes to developing future propulsion systems for interstellar travel, the principles of special relativity, including length contraction, will undoubtedly play a crucial role.

                        In conclusion, length contraction is a remarkable consequence of special relativity that challenges our intuitive understanding of space and time. By exploring this concept through derivations, examples, and real-world applications, we gain a deeper appreciation for the intricate nature of our universe. As we continue to unravel the mysteries of physics, length contraction serves as a reminder that reality can be far more complex and fascinating than it appears at first glance.

                        Relativistic Momentum

                        Relativistic momentum is a fundamental concept in special relativity that extends our understanding of classical momentum to account for objects moving at high speeds, approaching the speed of light. While classical momentum is simply the product of an object's mass and velocity (p = mv), relativistic momentum introduces important modifications to this formula.

                        In classical physics, momentum is a conserved quantity that remains constant in closed systems. However, as objects approach the speed of light, classical mechanics breaks down, and we must turn to relativistic mechanics. The key difference between classical and relativistic momentum lies in how they behave at high velocities.

                        The equation for relativistic momentum is given by:

                        p = γmv

                        Where:

                        • p is the relativistic momentum
                        • m is the rest mass of the object
                        • v is the velocity of the object
                        • γ (gamma) is the Lorentz factor

                        The Lorentz factor, γ, is a crucial component of relativistic calculations and is defined as:

                        γ = 1 / (1 - v²/c²)

                        Where c is the speed of light in a vacuum. As an object's velocity approaches the speed of light, the Lorentz factor increases, approaching infinity.

                        To understand relativistic momentum, it's essential to grasp the concept of relativistic mass. Relativistic mass is the apparent increase in an object's mass as its velocity increases. It's related to the rest mass by the equation:

                        m_relativistic = γm_rest

                        This relationship between relativistic mass and the Lorentz factor explains why the relativistic momentum equation differs from its classical counterpart. As an object's speed increases, its relativistic mass increases, leading to a greater increase in momentum than classical physics would predict.

                        To calculate relativistic momentum, follow these steps:

                        1. Determine the object's rest mass (m) and velocity (v).
                        2. Calculate the Lorentz factor (γ) using the equation γ = 1 / (1 - v²/c²).
                        3. Multiply the rest mass by the Lorentz factor to find the relativistic mass.
                        4. Multiply the relativistic mass by the velocity to obtain the relativistic momentum.

                        It's important to note that as an object's velocity approaches the speed of light, its relativistic momentum approaches infinity. This explains why it's impossible for massive objects to reach the speed of light it would require an infinite amount of energy to accelerate them to that speed.

                        The implications of relativistic momentum are profound in particle physics and astrophysics. For example, in particle accelerators, where subatomic particles are accelerated to speeds very close to the speed of light, relativistic effects become significant. The increased momentum of these particles allows for high-energy collisions that reveal fundamental properties of matter and energy.

                        In conclusion, relativistic momentum is a crucial concept that extends our understanding of motion and energy beyond the limitations of classical physics. By incorporating the Lorentz factor and the concept of relativistic mass, it provides a more accurate description of the behavior of objects moving at high speeds. This understanding is essential for advanced physics research and has practical applications in fields ranging from particle physics to space exploration.

                        Mass and Energy in Special Relativity

                        The relationship between mass and energy is one of the most profound discoveries in modern physics, fundamentally changing our understanding of the universe. At the heart of this relationship lies Albert Einstein's famous equation, E=mc^2, which forms the cornerstone of special relativity. This equation, simple in appearance yet revolutionary in its implications, establishes the principle of mass-energy equivalence.

                        E=mc^2 states that energy (E) is equal to mass (m) multiplied by the speed of light (c) squared. The speed of light, approximately 3 x 10^8 meters per second, is an enormous number. When squared, it becomes even more massive. This means that a small amount of mass can be converted into an enormous amount of energy, and conversely, a large amount of energy can be converted into a small amount of mass.

                        The concept of rest energy emerges from this equation. Even when an object is at rest, it possesses energy by virtue of its mass. This rest energy is given by E = mc^2, where E represents the rest energy. It's important to note that this energy is intrinsic to the object and is not related to its motion or position.

                        In relativistic terms, the total energy of an object includes both its rest energy and its kinetic energy. The total energy is expressed as E = γmc^2, where γ (gamma) is the Lorentz factor, given by 1/(1-v^2/c^2). Here, v represents the object's velocity. As an object's speed approaches the speed of light, γ increases, causing the total energy to grow dramatically.

                        Kinetic energy in special relativity takes on a new form compared to classical mechanics. The relativistic kinetic energy is given by K = (γ-1)mc^2. This equation shows that as an object's speed approaches the speed of light, its kinetic energy increases without bound. This explains why it's impossible for massive objects to reach the speed of light it would require infinite energy.

                        The implications of mass-energy equivalence are far-reaching. It explains the source of energy in nuclear reactions, both fission and fusion. In these processes, a small difference in mass between the reactants and products is converted into an enormous amount of energy. This principle powers stars, nuclear weapons, and nuclear power plants.

                        Moreover, mass-energy equivalence has profound consequences for our understanding of particle physics. It allows for the creation of particle-antiparticle pairs from pure energy and the annihilation of such pairs back into energy. This process is observed in high-energy particle colliders and plays a crucial role in our understanding of the early universe.

                        The equation E=mc^2 also impacts our understanding of gravity. In general relativity, Einstein's theory of gravity, energy and mass are treated as equivalent sources of gravitational fields. This means that not only does matter curve spacetime, but energy does as well, leading to phenomena like gravitational lensing by massive objects.

                        In conclusion, the relationship between mass and energy in special relativity, encapsulated in the equation E=mc^2, represents a fundamental shift in our understanding of the physical world. It unifies two seemingly distinct concepts mass and energy into a single, interchangeable entity. This principle underlies much of modern physics, from the workings of stars to the behavior of subatomic particles, and continues to shape our exploration of the universe.

                        Relativistic Addition of Velocities

                        In classical mechanics, velocities can be simply added together. For instance, if you're walking at 5 km/h on a train moving at 100 km/h, your velocity relative to the ground would be 105 km/h. However, in special relativity, this straightforward addition doesn't work. The reason lies in the fundamental principle that the speed of light is constant for all observers, regardless of their relative motion.

                        Special relativity introduces the relativistic velocity addition formula to accurately describe how velocities combine at speeds approaching that of light. This formula is:

                        v = (u + v') / (1 + uv'/c²)

                        Where:

                        • v is the relative velocity between the two objects
                        • u is the velocity of the first object in the observer's frame
                        • v' is the velocity of the second object relative to the first object
                        • c is the speed of light

                        To illustrate this concept, let's consider a basketball example. Imagine you're on a spaceship moving at 0.6c (60% the speed of light) relative to Earth. You throw a basketball forward at 0.5c relative to your spaceship. In classical mechanics, the ball's speed relative to Earth would be 1.1c, exceeding the speed of light. However, using the relativistic formula:

                        v = (0.6c + 0.5c) / (1 + 0.6 × 0.5) = 0.85c

                        The basketball's actual speed relative to Earth is 0.85c, which is less than the speed of light, adhering to the principles of special relativity.

                        To apply the relativistic velocity addition formula in different scenarios, follow these steps:

                        1. Identify the two velocities you're combining (u and v').
                        2. Express these velocities as fractions of the speed of light (c).
                        3. Plug the values into the formula: v = (u + v') / (1 + uv'/c²).
                        4. Simplify and solve for v.
                        5. Interpret the result, remembering that it will always be less than c.

                        This formula has interesting implications. As velocities approach the speed of light, the denominator grows, causing the combined velocity to approach, but never reach or exceed, c. This mathematical behavior ensures that the speed of light remains the universal speed limit.

                        In everyday life, where velocities are much smaller than c, the relativistic formula closely approximates classical velocity addition. The differences become significant only at extremely high speeds. For instance, if two spaceships approach each other, each moving at 0.9c relative to a stationary observer, their relative velocity is not 1.8c, but about 0.995c.

                        Understanding relativistic velocity addition is crucial in particle physics, where subatomic particles often move at speeds close to that of light. It's also essential in space travel calculations and in comprehending the fundamental nature of space and time as described by Einstein's special relativity.

                        In conclusion, the relativistic addition of velocities represents a significant departure from classical mechanics. It preserves the constancy of the speed of light and provides a more accurate description of motion at high speeds. By using the relativistic velocity addition formula, we can correctly calculate velocities in scenarios where classical physics breaks down, ensuring our understanding of the universe remains consistent with observed phenomena.

                        Applications and Implications of Special Relativity

                        Special relativity, a cornerstone of modern physics, has profound implications that extend far beyond theoretical discussions in academic circles. This revolutionary theory, proposed by Albert Einstein in 1905, has found numerous practical applications in our everyday lives and continues to shape our understanding of the universe.

                        One of the most ubiquitous applications of special relativity is in GPS systems. These navigation tools, which we rely on daily for accurate positioning and timing, wouldn't function correctly without accounting for relativistic effects. GPS satellites orbiting Earth move at high speeds relative to users on the ground, causing time to pass slightly slower for the satellites due to time dilation. Additionally, the weaker gravitational field experienced by the satellites results in time moving faster for them compared to Earth's surface. These tiny time differences, predicted by special and general relativity, accumulate rapidly. Without correcting for these effects, GPS systems would accumulate errors of about 10 kilometers per day, rendering them useless for precise navigation.

                        Particle accelerators, such as the Large Hadron Collider (LHC) at CERN, represent another area where special relativity plays a crucial role. These massive machines accelerate subatomic particles to speeds approaching that of light. As particles reach these extreme velocities, relativistic effects become significant. The increase in mass predicted by special relativity must be accounted for in the design and operation of these accelerators. Without considering relativistic mass increase, it would be impossible to accurately control and collide particles at such high energies. These experiments have led to groundbreaking discoveries in particle physics, including the confirmation of the Higgs boson in 2012.

                        In the realm of cosmology, special relativity forms the foundation for our modern understanding of the universe. The theory's prediction that the speed of light is constant in all reference frames has far-reaching consequences for how we perceive distant cosmic events. For instance, when observing supernovae or other celestial phenomena billions of light-years away, we must consider that we're seeing these events as they occurred in the distant past. This realization has revolutionized our concept of cosmic time and space, leading to the development of the broader theory of general relativity and our current model of an expanding universe.

                        The concept of space-time, a fundamental aspect of special relativity, has dramatically altered our view of the universe. Instead of treating space and time as separate entities, we now understand them as interconnected dimensions of a single continuum. This paradigm shift has profound implications for our understanding of causality, simultaneity, and the nature of reality itself. It challenges our intuitive notions of absolute time and has led to mind-bending concepts like time dilation and length contraction.

                        Beyond these scientific applications, special relativity has practical implications in various technological fields. For instance, in the development of atomic clocks, which are the most precise timekeeping devices available, relativistic effects must be considered to achieve unprecedented accuracy. These ultra-precise clocks are essential for synchronizing global communications networks, financial transactions, and scientific experiments.

                        The theory also finds applications in nuclear energy production. The famous equation E=mc², derived from special relativity, explains the enormous energy released in nuclear reactions. This understanding has led to the development of nuclear power plants and, unfortunately, nuclear weapons, showcasing both the beneficial and potentially destructive applications of scientific knowledge.

                        In conclusion, special relativity, while often perceived as an abstract theory, has profound and tangible impacts on our daily lives and our understanding of the cosmos. From the GPS systems guiding our travels to the particle accelerators probing the fundamental nature of matter, and from our evolving comprehension of the universe to the development of cutting-edge technologies, special relativity continues to shape our world in remarkable ways. As we continue to explore and apply these concepts, we can expect even more exciting discoveries and innovations that will further transform our relationship with space, time, and the universe at large.

                        Conclusion

                        In this article, we've explored key concepts of special relativity, including length contraction, relativistic momentum, mass-energy equivalence, and velocity addition. These complex ideas challenge our everyday understanding of physics. The introduction video serves as a crucial foundation for grasping these concepts, providing visual aids and explanations that make the abstract more tangible. To truly master these principles, it's essential to engage further with the material. We encourage readers to tackle practice problems in relativity, which can solidify understanding and reveal nuances in the theory. Additionally, exploring supplementary resources, such as advanced textbooks or online lectures, can provide deeper insights into the fascinating world of special relativity. Remember, comprehending these concepts takes time and effort, but the rewards are immense, offering a profound view of the universe's fundamental workings. Continue your journey into special relativity, and don't hesitate to revisit this article and its accompanying video as you progress in your studies.

                        Length Contraction & Relativistic Momentum, Mass and Energy

                        Introduction to length contraction

                        Step 1: Understanding Length Contraction

                        Length contraction is a phenomenon in special relativity where the length of an object moving at a significant fraction of the speed of light appears shorter along the direction of motion when observed from a stationary frame of reference. This concept is similar to time dilation, where time intervals differ between moving and stationary observers. The key idea is that as an object's speed increases, its length contracts in the direction of motion.

                        Step 2: The Effect of Speed on Length

                        To understand how speed affects the length of an object, consider a spaceship traveling at high speed from Earth to another planet. We need to analyze the situation from two different frames of reference: one from an observer on Earth and another from an observer inside the spaceship. The observer on Earth measures the distance between the planets (L0) and the time required to cover this distance (Δt). The time is calculated using the formula Δt = L0 / V, where V is the velocity of the spaceship.

                        Step 3: Observations from the Spaceship

                        For the observer inside the spaceship, the situation appears different. As the spaceship moves towards the planet, the observer inside feels that the planet is moving towards them, and the Earth is moving away. The time interval for the observer inside the spaceship is shorter due to the effects of time dilation. The time dilation equation is Δt0 = Δt * sqrt(1 - V^2 / c^2), where c is the speed of light.

                        Step 4: Calculating Length Contraction

                        To find the contracted length (L) as viewed by the observer inside the spaceship, we use the relationship between distance, velocity, and time. The distance L is given by L = V * Δt0. Substituting Δt0 from the time dilation equation, we get L = V * Δt * sqrt(1 - V^2 / c^2). Since V * Δt = L0, we can simplify this to L = L0 * sqrt(1 - V^2 / c^2). This is the length contraction equation.

                        Step 5: Simplifying the Length Contraction Equation

                        We can further simplify the length contraction equation by introducing the Lorentz factor (γ), which is defined as γ = 1 / sqrt(1 - V^2 / c^2). Using this, the length contraction equation becomes L = L0 / γ. This shows that the length of an object moving at high speed is shorter by a factor of γ compared to its length when at rest.

                        Step 6: Practical Example of Length Contraction

                        Consider a rectangular painting with dimensions 1 meter by 1.5 meters hung on the wall of a spaceship. If the spaceship is moving along the x-axis, only the length along the direction of motion (1.5 meters) will be affected by length contraction. Using the length contraction equation, we can calculate the contracted length. If the spaceship's speed is 0.8c, the contracted length is L = 1.5 * sqrt(1 - 0.8^2) = 0.9 meters. The width (1 meter) remains unchanged as it is perpendicular to the direction of motion.

                        Step 7: Conclusion on Length Contraction

                        Length contraction occurs only along the direction of motion and is a direct consequence of the principles of special relativity. It demonstrates how measurements of space and time are relative to the observer's frame of reference. Understanding length contraction is crucial for comprehending the behavior of objects moving at relativistic speeds and the fundamental nature of space and time.

                        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.

                        Prerequisite Topics

                        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.

                        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)