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- If a charged particle like a proton is racing around a particle accelerator at nearly the speed of light, it gets shorter and heavier. Let's assume the gain in the mass of a speeding charge particle is result of motion.
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-. Now suppose it was impossible to measure “absolute motion“. Suppose all motion is “relative“. Suppose it all depends on your “frame of reference“.
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-. At this moment I am moving east at 700 miles per hour. So is my dog Molly. But we both are at rest unless we change our reference to account for how fast the Earth is rotating.
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-. Now it is logical to assume that velocities of motion add up. If a jet plane is going 700 miles per hour east and fires a rocket 700 miles per hour also due east the rocket should be traveling 1,400 miles per hour east. Right?
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------------------------ v = v1 + v2 = 700 + 700 = 1,400
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-. Now, if we change the “v” to the velocity of light which is “c“, and, we increase the velocity “v1” to near the speed of light, the velocities no longer add up to “v1” plus “v2“. In order to keep velocity constant when it reaches “c” we need to change the equation to read:
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------------------ v = ( v1 + v2 ) / ( 1 + v1*v2 / c^2 )
-. You can see how the equation works by letting “v1” equal to “c“.
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------------------- v = ( c + v2 ) / ( 1 + c*v2 / c^2 ) = ( c + v2 )* c^2 / ( c^2 + c*v2 ) = ( c + v2 ) *c / ( c + v2 ) = c
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- If you increase “v1” to “c” the whole expression reduces back to “c”, the speed of light is constant stop for velocity.
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-. These equations are symbols. The symbols say if you let one velocity reach the speed of light add another velocity to it , it still leaves you with the speed of light. You cannot escape it. You cannot go faster than the speed of light, according to this simple equation.
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-. Actually is something was going faster than the speed of light, it could happen, but we would never see it because the light would never reach us. We can never measure something going faster than the speed of light.
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-. So, maybe the correct way to say this is that we can never measure anything moving faster than speed of light.
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-. Now, once these equations held the speed of light constant other things happened to other equations.
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-. Not only charged particles gain mass with motion, everything does, even baseballs.
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-. Again, measuring mass is relative. In order to hold velocity constant the measured length of the object is shortened and the measured mass must increase.
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- Now we have arrived at another contradiction in science. “Mass increases“. The Conservation of Mass and Energy says neither can be created nor destroyed.
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-. Motion does not really create mass. The energy of motion is kinetic energy and kinetic energy equals one half mass times velocity squared.
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------------------------------ K E = ½ m*v^2
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-. Now, if we think of mass and energy being interchangeable, then thinking mass increases and energy decreases the combined mass -energy remains unchanged. Conservation is maintained. We can still abide by the Law of the Conservation of Mass- Energy.
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-. How much does mass change to energy?
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-. This answer gets right back to the velocity of light being constant. The time for a light beam to travel in any direction is the same.
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---------------------- t1 / t2 = 1 / square root ( 1 - v^2 / c^2 )
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-. This expression can be written using different math symbols.
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------------------------------ ( 1 - v^2 / c^2 ) ^ -½
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-. The minus sign “-” in the exponent means reciprocal
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-------------------------- 3 ^-1 = 1/3
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-. The one half, “ ½”, in the exponent means the square root.
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---------------------- 3^½ = square root of 3 = 1.732
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-The expression ( x + y) is a binomial, and expression with two numbers.
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- 5th graders recognize the expression:
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------------------- ( x + y ) ^2 = ( x + y ) * ( x + y ) = x^2 + xy + y^2
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-. Okay let's go to 12th grade.
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-------------------- ( x + y ) ^4 = ( x + y ) * ( x + y ) *( x + y ) * ( x + y ) = x^4 +4 x^3y + 6x^2 y^2+ 4 xy^3 + y^4
-. The binomial coefficients appear as entries in Pascal's triangle where each entry is the sum of two above:
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---------------------------------- 1
--------------------------------- 1 1
-------------------------------- 1 2 1
------------------------------- 1 3 3 1
------------------------------ 1 4 6 4 1
--------------------------- 1 5 10 10 5 1
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----------- note: ( x+ y)^4 having he coefficients 1 4 6 4 1
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-. Isaac Newton discovered the mathematical theorem called the “Binomial Theorem” where the binomial expression ( 1 - x) ^-n can be expanded into a endless series.
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-. BINOMIAL THEOREM:
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------------------ ( 1 - x) ^-n = 1 + nx + ( n(n+1) x^2 / 2!) + ( n(n+1)(n+2)x^3 / 3!) + ( n(n+1)(n+2)(n+3)x^4 / 4!) + ……………………………
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-. In the case of the binomial expression.
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------------------------------- ( 1 - v^2 / c^2 ) ^ -½
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------------------------------ “ n” = ½
------------------------------- “x” = v^2 / c^2
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-------- ( 1 - v^2 / c^2 ) ^ -½ = 1 + v^2 / 2 c^2 + 3 v^4 / 8 c^4 + 4v^5 / 24c^5 + ……………………………….........
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-. Once “v” increases to “c” that is, the velocity equals the speed of light, the binomial series coefficients become:
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---------------------- 1 + ½ 3/8 + 4/24 + 5/120 + ……………..
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-. When velocity slows to one half the speed of light.
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------------------- v = c/2
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-. The series becomes:
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------------------------------ 1 + 1/8 + 3/128 + 4/ 384 + 5/ 3,840
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-. When the velocity slows to one fourth the speed of light, v = c/4
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-. The series becomes:
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----------------------------- 1 + 1/32 + 3 / 2,048
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-. Therefore , at ordinary velocities a very small fraction of the speed of light the series terms after the first two become exceedingly small. The expression can be assumed to be nearly equal to the first two terms in the series.
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--------------------------- ( 1 - v^2 / c^2 ) ^ -½ = (1 + v^2 / 2 c^2 )
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- In the general sense this equation is expressed as ( 1+x^a ) = 1 + ax,…………….. …….. (approximately)
-. Applying this expression to the increase in mass
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---------------- m1 = m0 (1 + v^2 / 2 c^2 )
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--------------- m1 = m0 + ½ m v^2 / c^2
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-. The increase in mass is:
---------------- (m1 - m0) = (½ m v^2 / c^2)
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-. (½ m v^2 / c^2) is the energy of a moving body, Kinetic Energy. Let K.E. = “E”. Let the change in mass just equal mass “m” .
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-------------------- m = E / c^2
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------------------- E = mc^2
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-. Congratulations!
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-. You are a genius!
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-. You just derived the famous equation E = mc^2 using the binomial theorem. You can apply this same derivation for length shortening and time slowing. Everything is relative.
- -----------------------------------------------------------------------------------------------
RSVP, with comments, suggestions, corrections. Index of reviews available ---
--- Some reviews are at: -------------------- http://jdetrick.blogspot.com -----
---- email request for copies to: ------- jamesdetrick@comcast.net ---------
---- https://plus.google.com/u/0/ , “Jim Detrick” ----- www.facebook.com ---
---- www.twitter.com , --- 707-536-3272 ---- Wednesday, March 19, 2014 ---
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