Deriving Einstein's most famous equation： Why does energy mass x speed of light squared？ (copy)
The equation equals MC squared is the most famous equation in physics
Very few people know what the equation means or where it comes from
The video aims to derive the equation and provide insight into its meaning
Touch upon fascinating features of Einstein's theory of special relativity
Mechanics describes how bodies change position in space with respect to time
Observing the motion of a stone dropped from a moving train
Stone descends in a straight line relative to the train
Stone falls in a parabolic curve relative to the ground
Introduction of the concept of motion relative to a system of coordinates
Definition of a frame of reference and the principle of inertia
Inertial frames of reference are non-accelerating frames
Introduction of the principle of relativity by Galileo
Discussion of Galileo's dialogue concerning the two chief world systems
Thought experiment on a ship to demonstrate the principle of relativity
No experiment can distinguish between a stationary frame and a moving frame
Assumption that the laws of physics are the same in all inertial frames
Explanation of the Galileian addition of velocities
Example of a person walking on a moving train
Calculation of the person's velocity relative to the ground
Application of the concept to the propagation of light
Questioning the reference frame for the speed of light
Analogous situation of a light beam on a moving train
Conclusion that the speed of light is relative to the track
Einstein's realization about the constancy of the speed of light
Explanation of how a person standing still and a person moving can measure the same speed of light
Introduction of a simple kind of clock consisting of parallel mirrors
Comparison of a stationary clock and a moving clock
Derivation of the time taken for a tick of the moving clock
Introduction of the factor gamma to quantify the difference in time between the moving clock and the stationary clock
Explanation that any type of clock would run slow by the same amount if the principle of relativity is correct
The speed of light appears to conflict with the constancy of the speed of light implied by Maxwell's equations
Einstein realized that the only way for a person standing still and a person moving to measure the same speed of light is if their sense of space and time was not the same
Relative to the spaceship, the light is traveling 300,000 spaceship kilometers per spaceship second
Relative to the road, the light is traveling 300,000 road kilometers per road second
The clock consists of 2 perfectly parallel mirrors separated by one meter
A light signal is sent between the two ends, making a tick every time it moves up and a talk every time it comes down
Two clocks with the same length are synchronized and agree afterwards
The time taken for the tick tock of the stationary clock is denoted as T Naught
The time taken for the tick tock of the moving clock is denoted as t
The path taken by the light during the tick tock of the moving clock is longer due to the motion of the train
The time taken for a tick of the moving clock is calculated as 2 divided by the square root of c squared minus v squared
The expression for the time taken for a tick of the moving clock is written as t equals gamma T naught
Gamma is the factor that tells us how significant the difference is between T and T naught
If v equals 0, then gamma is equal to 1 and the clocks tick and talk at the same rate
If v is greater than 0, then gamma will be greater than 1 and the time taken for the tick tock of the moving clock will be greater than the time taken for the tick tock of the stationary clock
Mismatch between clocks on a moving train violates the principle of relativity
All moving clocks run slower by the same amount
Time appears to be slower in the moving train
Time dilation is the difference in elapsed time between stationary and moving frames of reference
As the speed of an object approaches the speed of light, time slows down
The tick tock of a moving clock gets slower until time stands still
Slowing of time is not noticeable in day-to-day life for most moving objects
Example: Usain Bolt running at an average speed of 10 meters per second does not experience noticeable time dilation
Muons are fundamental particles created by cosmic radiation in the upper atmosphere
Muons have a short average lifetime of 2.2 microseconds
Despite their short lifetime, muons can travel over 10,000 meters due to time dilation
Time dilation allows muons to be detected in laboratories on the surface of the Earth
Relativistic energy is determined by considering the gain in kinetic energy of an object
Work done on an object is equal to the integral of force multiplied by displacement
Force can be expressed as the rate of change of momentum
Momentum is defined as the mass of an object multiplied by its velocity
Relativistic momentum is given by gamma times mass times velocity
Einstein's equation, E = mc^2, is linked to the relativistic energy of an object
Applying the product rule, we find the following ghastly green expression.
Calculating the highlighted derivative of 1 minus v squared over c squared raised to the power of minus 1 half.
We find the following blue expression which simplifies to Vovercsquareddvbydt multiplied by 1 minus v squared over c squared raised the power of minus 3 over 2.
Derivative of the momentum
Both terms contain M times DV by DT, which can be factored out.
Arriving at the following green expression.
Writing the expression as a single fraction with denominator 1 minus v squared over c squared raised the power 3 over 2.
The v squared over c squared terms in the numerator cancel.
Arriving at the simplified pink expression, which is equal to gamma cubed MDV by DT.
Substituting the derivative of the momentum back into the integral to determine the equation for the kinetic energy.
Combining all calculated expressions to find the expression for the kinetic energy.
Changing variables to integrate with respect to V instead of X.
Writing DV by DT times DX as DX by DT times DV.
Changing the limits of the integral.
Integrating with respect to V and substituting the limit to arrive at the pink expression.
Analyzing the equation for the kinetic energy
Writing the equation in a more compact form as MC squared gamma-1.
Setting V equals to 0 to find that the kinetic energy is equal to 0 for a stationary object.
Understanding the equation as the total energy of the object, denoted as e.
Interpreting the total energy equation
Writing the total energy as MCsquared times 1 minus v squared over c squared to the power of minus 1 half.
Expanding the equation in the low velocity limit to find the classical equation for kinetic energy.
Recognizing MC squared as the rest mass energy, which exists even when the object is at rest.
Understanding the rest mass energy
Considering the rest energy as the energy required to create the mass of the object.
Noting that the rest energy does not depend on the velocity and is a different form of energy.
The more massive an object, the more energy is required to create that mass
Rest energy is a form of stored energy
Mass can be converted into energy through processes like nuclear fusion and annihilation
Einstein's famous equation, E=mc^2, relates energy and mass
The speed of light represents an upper limit to the velocity an object can possess
Massless particles, like photons, have energy but no mass
Energy can be expressed in terms of momentum
More energy is required to create a more massive object
Rest energy is a form of stored energy in the mass of an object
Stars produce energy through the conversion of mass into energy during nuclear fusion
Matter and anti-matter can annihilate each other, converting mass into energy
The energy released in annihilation can be calculated using E=mc^2
Annihilation of an electron and positron releases a significant amount of energy
Annihilating large amounts of matter and anti-matter could potentially produce massive amounts of energy
Anti-matter production is currently not feasible for mass-scale energy production on Earth
The equation E=mc^2 relates energy and mass
For a freely moving object, the total energy is given by E=γmc^2
The total energy consists of kinetic energy and rest energy
The rest energy equation is the most famous equation in physics
If an object is moving, the appropriate equation to use is E=γmz^2
The speed of light represents an upper limit to the velocity an object can possess
Increasing the velocity of a massive object towards the speed of light requires infinite energy
The speed of light is a fundamental limit to the velocity of any object
Photons are massless particles that travel at the speed of light
The energy of a photon can be calculated using E=γmc^2
When substituting m=0 and V=C, both the numerator and denominator become 0
This suggests that the energy of massless particles should not be thought of in terms of velocity
Massless particles can have different energies even if they travel at the same velocity
Energy can be expressed in terms of the momentum of a particle
Kinetic energy in classical physics: KE = 1/2mv^2
Relating kinetic energy and momentum: KE = P^2 / 2m
Relativistic energy equation: e = γmz^2
Squaring both sides of the equation: e^2 = m^2c^4 / (1 - v^2/c^2)
Combining terms: e^2 = m^2c^4 - m^2c^2v^2 / (1 - v^2/c^2)
Relativistic momentum: P = mγv
Energy equation in terms of momentum and mass: e^2 = P^2c^2 + m^2c^4
Energy equation for massless particles: e = Pc
Massless particles carry momentum: momentum = energy / speed of light
Momentum of a massless photon: momentum = Planck's constant / wavelength (λ)
Energy and momentum of an isolated system are conserved
Observers assign different values for energy and momentum, but mass remains invariant
Mass is a fundamental invariant of the theory of relativity
Particle physicists determine masses of fundamental particles using the energy-momentum relation
The Higgs Boson discovery and mass determination in 2012
Curiosity and questioning are important
Comprehending the mysteries of reality a little each day
Thank you for watching.
The equation equals MC squared is the most famous equation in physics
Very few people know what the equation means or where it comes from
The video aims to derive the equation and provide insight into its meaning
Touch upon fascinating features of Einstein's theory of special relativity
Mechanics describes how bodies change position in space with respect to time
Observing the motion of a stone dropped from a moving train
Stone descends in a straight line relative to the train
Stone falls in a parabolic curve relative to the ground
Introduction of the concept of motion relative to a system of coordinates
Definition of a frame of reference and the principle of inertia
Inertial frames of reference are non-accelerating frames
Introduction of the principle of relativity by Galileo
Discussion of Galileo's dialogue concerning the two chief world systems
Thought experiment on a ship to demonstrate the principle of relativity
No experiment can distinguish between a stationary frame and a moving frame
Assumption that the laws of physics are the same in all inertial frames
Explanation of the Galileian addition of velocities
Example of a person walking on a moving train
Calculation of the person's velocity relative to the ground
Application of the concept to the propagation of light
Questioning the reference frame for the speed of light
Analogous situation of a light beam on a moving train
Conclusion that the speed of light is relative to the track
Einstein's realization about the constancy of the speed of light
Explanation of how a person standing still and a person moving can measure the same speed of light
Introduction of a simple kind of clock consisting of parallel mirrors
Comparison of a stationary clock and a moving clock
Derivation of the time taken for a tick of the moving clock
Introduction of the factor gamma to quantify the difference in time between the moving clock and the stationary clock
Explanation that any type of clock would run slow by the same amount if the principle of relativity is correct
The speed of light appears to conflict with the constancy of the speed of light implied by Maxwell's equations
Einstein realized that the only way for a person standing still and a person moving to measure the same speed of light is if their sense of space and time was not the same
Relative to the spaceship, the light is traveling 300,000 spaceship kilometers per spaceship second
Relative to the road, the light is traveling 300,000 road kilometers per road second
The clock consists of 2 perfectly parallel mirrors separated by one meter
A light signal is sent between the two ends, making a tick every time it moves up and a talk every time it comes down
Two clocks with the same length are synchronized and agree afterwards
The time taken for the tick tock of the stationary clock is denoted as T Naught
The time taken for the tick tock of the moving clock is denoted as t
The path taken by the light during the tick tock of the moving clock is longer due to the motion of the train
The time taken for a tick of the moving clock is calculated as 2 divided by the square root of c squared minus v squared
The expression for the time taken for a tick of the moving clock is written as t equals gamma T naught
Gamma is the factor that tells us how significant the difference is between T and T naught
If v equals 0, then gamma is equal to 1 and the clocks tick and talk at the same rate
If v is greater than 0, then gamma will be greater than 1 and the time taken for the tick tock of the moving clock will be greater than the time taken for the tick tock of the stationary clock
Mismatch between clocks on a moving train violates the principle of relativity
All moving clocks run slower by the same amount
Time appears to be slower in the moving train
Time dilation is the difference in elapsed time between stationary and moving frames of reference
As the speed of an object approaches the speed of light, time slows down
The tick tock of a moving clock gets slower until time stands still
Slowing of time is not noticeable in day-to-day life for most moving objects
Example: Usain Bolt running at an average speed of 10 meters per second does not experience noticeable time dilation
Muons are fundamental particles created by cosmic radiation in the upper atmosphere
Muons have a short average lifetime of 2.2 microseconds
Despite their short lifetime, muons can travel over 10,000 meters due to time dilation
Time dilation allows muons to be detected in laboratories on the surface of the Earth
Relativistic energy is determined by considering the gain in kinetic energy of an object
Work done on an object is equal to the integral of force multiplied by displacement
Force can be expressed as the rate of change of momentum
Momentum is defined as the mass of an object multiplied by its velocity
Relativistic momentum is given by gamma times mass times velocity
Einstein's equation, E = mc^2, is linked to the relativistic energy of an object
Applying the product rule, we find the following ghastly green expression.
Calculating the highlighted derivative of 1 minus v squared over c squared raised to the power of minus 1 half.
We find the following blue expression which simplifies to Vovercsquareddvbydt multiplied by 1 minus v squared over c squared raised the power of minus 3 over 2.
Derivative of the momentum
Both terms contain M times DV by DT, which can be factored out.
Arriving at the following green expression.
Writing the expression as a single fraction with denominator 1 minus v squared over c squared raised the power 3 over 2.
The v squared over c squared terms in the numerator cancel.
Arriving at the simplified pink expression, which is equal to gamma cubed MDV by DT.
Substituting the derivative of the momentum back into the integral to determine the equation for the kinetic energy.
Combining all calculated expressions to find the expression for the kinetic energy.
Changing variables to integrate with respect to V instead of X.
Writing DV by DT times DX as DX by DT times DV.
Changing the limits of the integral.
Integrating with respect to V and substituting the limit to arrive at the pink expression.
Analyzing the equation for the kinetic energy
Writing the equation in a more compact form as MC squared gamma-1.
Setting V equals to 0 to find that the kinetic energy is equal to 0 for a stationary object.
Understanding the equation as the total energy of the object, denoted as e.
Interpreting the total energy equation
Writing the total energy as MCsquared times 1 minus v squared over c squared to the power of minus 1 half.
Expanding the equation in the low velocity limit to find the classical equation for kinetic energy.
Recognizing MC squared as the rest mass energy, which exists even when the object is at rest.
Understanding the rest mass energy
Considering the rest energy as the energy required to create the mass of the object.
Noting that the rest energy does not depend on the velocity and is a different form of energy.
The more massive an object, the more energy is required to create that mass
Rest energy is a form of stored energy
Mass can be converted into energy through processes like nuclear fusion and annihilation
Einstein's famous equation, E=mc^2, relates energy and mass
The speed of light represents an upper limit to the velocity an object can possess
Massless particles, like photons, have energy but no mass
Energy can be expressed in terms of momentum
More energy is required to create a more massive object
Rest energy is a form of stored energy in the mass of an object
Stars produce energy through the conversion of mass into energy during nuclear fusion
Matter and anti-matter can annihilate each other, converting mass into energy
The energy released in annihilation can be calculated using E=mc^2
Annihilation of an electron and positron releases a significant amount of energy
Annihilating large amounts of matter and anti-matter could potentially produce massive amounts of energy
Anti-matter production is currently not feasible for mass-scale energy production on Earth
The equation E=mc^2 relates energy and mass
For a freely moving object, the total energy is given by E=γmc^2
The total energy consists of kinetic energy and rest energy
The rest energy equation is the most famous equation in physics
If an object is moving, the appropriate equation to use is E=γmz^2
The speed of light represents an upper limit to the velocity an object can possess
Increasing the velocity of a massive object towards the speed of light requires infinite energy
The speed of light is a fundamental limit to the velocity of any object
Photons are massless particles that travel at the speed of light
The energy of a photon can be calculated using E=γmc^2
When substituting m=0 and V=C, both the numerator and denominator become 0
This suggests that the energy of massless particles should not be thought of in terms of velocity
Massless particles can have different energies even if they travel at the same velocity
Energy can be expressed in terms of the momentum of a particle
Kinetic energy in classical physics: KE = 1/2mv^2
Relating kinetic energy and momentum: KE = P^2 / 2m
Relativistic energy equation: e = γmz^2
Squaring both sides of the equation: e^2 = m^2c^4 / (1 - v^2/c^2)
Combining terms: e^2 = m^2c^4 - m^2c^2v^2 / (1 - v^2/c^2)
Relativistic momentum: P = mγv
Energy equation in terms of momentum and mass: e^2 = P^2c^2 + m^2c^4
Energy equation for massless particles: e = Pc
Massless particles carry momentum: momentum = energy / speed of light
Momentum of a massless photon: momentum = Planck's constant / wavelength (λ)
Energy and momentum of an isolated system are conserved
Observers assign different values for energy and momentum, but mass remains invariant
Mass is a fundamental invariant of the theory of relativity
Particle physicists determine masses of fundamental particles using the energy-momentum relation
The Higgs Boson discovery and mass determination in 2012
Curiosity and questioning are important
Comprehending the mysteries of reality a little each day
Thank you for watching.
