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# SRnotes - Intermediate Modern Physics - Review Problems...

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Intermediate Modern Physics - Review Problems Fall, 2009 1. A proton moving with β = 0 . 99 will appear like a moving disk. (a) Compute the thickness of the disk assuming that in the proton’s rest frame the proton’s diameter is 1 fm. (b) Compute the proton’s kinetic energy in MeV, assuming its mass is 938 MeV/c 2 . The proton collides and fuses with a stationary proton to create a new particle. (c) What is the mass of the new particle? (d) How fast will the new particle be moving? solution (a) The proton will be Lorentz-contracted by the factor γ = 1 / p 1 - β 2 = 7 . 09, so its thickness is 1fm / 7.09 = 0.14 fm. (b) According to Einstein, the total energy of a (free) particle is E = γmc 2 and mc 2 is the rest energy. The kinetic energy is therefore K = E - mc 2 = ( γ - 1) mc 2 . Given γ = 7 . 09 and mc 2 = 938 MeV, we ﬁnd K = 5 . 7 GeV. (c) Since 4-momentum is conserved, we can write P 1 + P 2 = P 3 , where P 1 and P 2 are the 4-momenta of the incident and stationary protons, respectively, and P 3 is the 4-momentum of the new particle. The rest energy of the new particle, and therefore its mass, is given by P 2 3 = ( m 3 c 2 ) 2 1 , that is, by P 2 3 = P 2 1 + P 2 2 + 2 P 1 P 2 , = 2( mc 2 ) 2 + 2 P 1 P 2 , = 2( mc 2 ) 2 + 2( γmc 2 , ~ pc )( mc 2 , ~ 0) , = 2( mc 2 ) 2 (1 + γ ) . (1) Therefore, m 3 c 2 = mc 2 p 2(1 + γ ) = 3 . 77 GeV. (d) The total energy of the new particle is E 3 = E 1 + E 2 = mc 2 ( γ + 1), so its gamma- factor is γ 3 = E 3 /m 3 c 2 ; that is, mc 2 (1 + γ ) /mc 2 p 2(1 + γ ) = p (1 + γ ) / 2 = 2 . 01. Therefore, β 3 = p γ 2 3 - 1 3 = 0 . 87. 1 Since P = ( γmc 2 , ~ pc ), we have that P 2 = γ 2 ( mc 2 ) 2 - ( γmvc ) 2 , that is, P 2 = γ 2 ( mc 2 ) 2 (1 - β 2 ) = ( mc 2 ) 2 . 1

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2. Compute the spacetime interval (in light years ) between the following events A and B , with the time and space coordinates given in Earth’s frame of reference: A — time 2019 AD on Earth B — time 2025 AD at Sirius, which is 8 light years away. What is the physical signiﬁcance of this time interval? solution The spatial separation is x = 8 ly and the temporal separation is ct = c (2025 - 2019) = 6 ly. Therefore, since
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## This note was uploaded on 02/17/2010 for the course PHY modern phy taught by Professor Prosper during the Fall '09 term at FSU.

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SRnotes - Intermediate Modern Physics - Review Problems...

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