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# sol14 - HW Solutions 14 8.01 MIT Prof Kowalski Relativity 1...

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HW Solutions # 14 - 8.01 MIT - Prof. Kowalski Relativity . 1 ) 37.20 Relativistic Relative Velocity Write differential form of Lorentz transformation: x = γ ( x ut ) t = γ ( t ux/c 2 ) 1 γ = 1 u 2 /c 2 which is: dx = γ ( dx udt ) (1) dt = γ ( dt udx/c 2 ) (2) Divide (2) by (1) you’ll get ( dx/dt = v x ): v = v x u x 1 uv x /c 2 (3) Where u is the velocity of the moving frame. In this problem you want to know the relative velocity so it means that your moving frame has the velocity of one of them and in this frame you want to know velocity of the other: v x = 0 . 9520 c u = 0 . 9520 c Replace this into equation (3) you’ll get: v = 0 . 9520 c ( 0 . 9520 c ) = 0 . 9988 c x 1 (0 . 9520 c )( 0 . 9520 c ) /c 2 v . 9988 c rel = 0 1

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2 ) 37.44 Creating a Particle a ) The total energy of a particle is: E = mγc 2 Where m is the rest mass and 1 γ = (4) 2 /c 2 1 u Before collision - incident two protons -: E = E P + E P = 2 m P γ P c 2 After collision - two protons + new particle which are at rest -: E = E P + E P + E η 0 = (2 m P + m η 0 ) c 2 From the energy conservation
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