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----------------------------- 2701 - RELATIVITY - as best I can explain it?
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- So you want to learn how Relativity works and how to derive Einstein’s calculations of its effect using the Pythagorean Theorem and simple algebra. Remember, the Pythagorean Theorem is the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. The equation has been around since 530 years before Christ.
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- Relativity tells us that if an objects motion becomes faster and faster, Time slows down, Mass increases, and space, or Length in the direction of motion shortens. This always happens with any object in motion except we do not notice it until motion approaches the speed of light, which is a constant 3*30^8 meters per second.
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- As we approach this velocity of 670,000,000 miles per hour Mass increases to infinity and the energy, or force, required to accelerate it any faster also increases to infinity. (Force = mass * acceleration). That is why nothing can ever travel faster than light. The only thing traveling light speed is weightless, has no mass. If something is weightless, like the photon, it must always and instantly travel at the speed of light.
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- At the same time Mass is increasing Time is slowing down. At the speed of light time stops. Photons do not experience time. At the same time space is shortening, the Length of an object in the direction of motion is getting shorter. At the speed of light length is reduced to zero.
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- If you can remember these basic principles of Relativity, Time slows, Length shortens, Mass increases, with increasing velocity, then , you can derive all the equations for Relativity.
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- In these calculations we will use the ratio, “v/c” which is the velocity of the object over the speed of light. We will use as our example a velocity of 90% the speed of light, which is 60,300,000 miles per hour, or, 17,000 miles per second. “v/c” = 0.9.
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- Distance is equal to velocity * time (v*t). So, if we make a right triangle and we make the hypotenuse the distance of “c * t“, the speed of light times time. And, we make the adjacent side the velocity of the object times time, “v * t”. Then the opposite side is “c * Tr”, the speed of light times Relativity Time.
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- The hypotenuse and adjacent side portray the vectors in motion. The opposite side is shorter than the hypotenuse so that must be Relativity Time because Relativity Time is shorter. Once we have defined the three sides of the right triangle we can use the Pythagorean theorem to set up our equations.
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--------- (hypotenuse)^2 = (adjacent side)^2 + (opposite side)^2
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---------- (c*t)^2 = (v*t)^2 + (c*Tr)^2
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--------- (c*Tr)^2 = (c*t)^2 - (v*t)^2
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---------- Tr^2 = t^2 - (v/c)^2*t^2
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----------- Tr^2 = t^2 (1-v^2/c^2)
----------------------------------------- where v/c = 90%
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----------- t = 1 hour that passes on the spacecraft
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---------- Tr = Relativity Time that passes more slowly due to Time Dilation
---------- Tr^2 = 1 ( 1 - .9^2) = 1 - .81 = 0.19
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---------- Tr = 0.44 hours
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---------- Tr = 26 minutes.
---------- Time Dilation has slowed time down from 1 hour to 26 minutes due to the relative motion at 90% the speed of light.
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- The same technique can be used for Length shortening and Mass increasing. To calculate Relativity Length, “Lr”, we create the same right triangle however this time we label the hypotenuse “c*l”. The adjacent side “v*l”. And, the opposite side “c*Lr”. We know the “c*Lr” has to be on the opposite side, and shorter side, because Length shortens due to Relativity.
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---------- (c*l)^2 = (v*l)^2 + (c*Lr)^2
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--------- (c*Lr)^2 = (c*l)^2 - (v*l)^2
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---------- Lr^2 = l^2 - (v/c)^2*l^2
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---------- Tr^2 = l^2 (1-v^2/c^2)
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------------------------------------- where v/c = 90%
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----------- l = 100 meters, the length of the spacecraft at rest
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---------- Lr = the length of the space craft in Relative motion.
--------- Lr^2 = 100* ( 1 - .9^2) = 100*(1 - .81) = 100*(.19) = 1,900 meters^2
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---------- Lr = 44 meters
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- The 100 meter spacecraft is shortened to 44 meters due to its relative speed that is 90% the speed of light.
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- The same technique is used to calculate the increase in Mass. This time we know that mass increases with velocity therefore Relativity Mass must be the term on the hypotenuse, the longer side. The hypotenuse has the term, “c*Mr”. The adjacent side also has to have the “ Mr” term, “v*Mr” And, the opposite side the term, “c*m”. This changes the equation for the triangle so the Mass will be calculated larger, as required by Relativity
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---------- (c*Mr)^2 = (v*Mr)^2 + (c*m)^2
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---------- Mr^2 = (v/c)^2*Mr^2 + m^2
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---------- Mr^2 - (v/c)^2*Mr^2 = m^2
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----------- Mr^2 = m^2 / (1-v^2/c^2)
---------------------------------------------------- where v/c = 90%
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----------- m = the mass of the spacecraft is 100 kilograms
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---------- Mr = the increased mass due to the spacecraft traveling at 90% light speed.
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---------- Mr^2 = 100 ^2 / ( 1 - .9^2) = 100^2 / (1 - .81) = 100^2 / (.19)
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---------- Mr^2 = 52,632 kilograms^2
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---------- Mr = 229 kilograms.
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- The spacecraft has gained mass from 100 to 229 kilograms due to its relative speed at 90% the speed of light.
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- This simple technique with the right triangles will allow you to calculate the effects of Relativity on time, space, and mass. With the binomial theorem you can carry this technique one step further and derive the equation for E = mc^2.
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- We start with our last expression for the mass with Relativity:
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----------- Mr^2 = m^2 / (1-v^2/c^2)
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----------- Mr = m / (1-v^2/c^2)^.5
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- Next, we recognize that the binomial term (1-v^2/c^2) ^.5 always appears in these formulas. This term is a simple binomial of the form ( x+ y)^n. The Binomial Theorem states that a binomial in this form can be expanded to a series:
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------------- (x + y)^n = x^n + x^(n-1) * y + x^(n-2) * y^2 + x^(n-3) * y^3 + ………
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- In the form (1+x)^-n this same binomial expansion becomes:
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------------- (1+x)^-n = 1 - n*x / 1! + n(n+1)x^2 / 2! - n(n+1)(n+2) x^3 / 3! + ……
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------------ where: x^2 is less than 1. It always is because “v” is always less than “c”
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------------ where: n = 0.5 for the square root
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----------- where x = (v^2/c^2)
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------------- (1+v^2/c^2)^-0.5 = 1 - 0.5 *(v^2/c^2)/ 1! + 0.5 (0.5 +1)(v^2/c^2)^2 / 2! - 0.5 (0.5 +1)(0.5 +2) (v^2/c^2)^3 / 3! + ……
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------------(1+v^2/c^2)^-0.5 = 1 - 1/2 *(v^2/c^2) + 0.375(v^4/c^4) - 0.3215 (v^6/c^6) + .2734 (v^8/c^8) + ………
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- You can see that after the first two terms the expression has c^4 and c^6 and c^8 in the denominator which is a very large number in the denominator so there terms become negligible. We are left with:
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----------- Mr = m * (1-v^2/c^2)-.5
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------------ Mr = = m - 1/2 *m*(v^2/c^2) + 0.375 m*(v^4/c^4) - 0.3215m* (v^6/c^6)
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------------ where; multiplying through by c^2:
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---------- Mr*c^2 = m*c^2 + ½ m*v^2 + …… negligible terms.
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------------ ½ m*v^2 is Kinetic Energy. The energy of mass in motion. But, there is another term which must be the energy when the energy when the mass is at rest, Energy at rest = mc^2. Einstein figured that this amount of energy existed in the Mass even when it was at rest.
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- In conclusion the energy of any object is equal to mc^2 plus its kinetic energy. We do not normally think of it that way because the kinetic energy is so small compared to mc^2 that we ignore it. However, as speeds approach the speed of light this term must be included.
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Interesting huh? Energy = mass * 90,000,000,000,000,000 even when mass is resting on the couch.
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- April 8, 2020 874 2701
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--------------------- Wednesday, April 8, 2020 -------------------------
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