Question

A proton in a high-energy accelerator moves with a speed of c/2. Use the work–kinetic energy theorem to find the work required to increase its speed to the following speeds.

A. .710c answer in units of MeV?

b..936c answer in units of Gev?

Answer 1

E = m c²

m = m_o / SQRT(1 - v²/c²)

E_o = m_o c²

KE = E - E_o

KE = m_o c² [ {1/SQRT(1 - v²/c²)} - 1]

E_o = m_o c²

E_o = (1.672 x 10 ^{- 27} kg)(3 x 10⁸ m/s)²

E_o = 5.016 x 10 ^{- 11} J

1 eV = 1.6 x 10 ^{- 19} J

1 MeV = 1.6 x 10 ^{- 16} J

Therefore,

E_o = 5.016 x 10 ^{- 11} J [1 MeV / 1.6 x 10 ^{- 16} J]

E_o = 313 500 Mev

E_o = 313.5 Gev

E = m c²

m = m_o / SQRT(1 - v²/c²)

E_orig = [m_o / SQRT(1 - v_orig²/c²)] c²

E_orig = [m_o c²][1/ SQRT(1 - v_orig²/c²)]

E_orig = [313.5 Gev][1/ SQRT(1 - {0.50 c}²/c²)]

E_orig = [313.5 Gev][1/ SQRT(1 - {0.50}²)]

E_orig = [313.5 Gev][1/ SQRT(1 - 0.25)]

E_orig = [313.5 Gev][1/ SQRT(0.75)]

E_orig = [313.5 Gev][1/ 0.866]

E_orig = [313.5 Gev][1.155]

E_orig = 362 Gev

E_a = [m_o / SQRT(1 - va²/c²)] c²

E_a = [m_o c²][1/ SQRT(1 - v_a²/c²)]

E_a = [313.5 Gev][1/ SQRT(1 - {0.710 c}²/c²)]

E_a = [313.5 Gev][1/ SQRT(1 - {0.710}²)]

E_a = [313.5 Gev][1/ SQRT(1 - 0.5041)]

E_a = [313.5 Gev][1/ SQRT(0.4959)]

E_a = 446 GeV

E_a = E_a - E_orig

E_a = (446 - 362) GeV

E_a = 84 GeV = 8.4000E+13 mev

E_b = [m_o / SQRT(1 - v_b²/c²)] c²

E_b = [m_o c²][1/ SQRT(1 - v_b²/c²)]

E_b = [313.5 Gev][1/ SQRT(1 - {0.936 c}²/c²)]

E_b = [313.5 Gev][1/ SQRT(1 - {0.936}²)]

E_b = [313.5 Gev][1/ SQRT(1 - 0.876)]

E_b = 890GeV

E_b = E_b - E_orig

  E_b = (890 - 362) GeV

E_b = 528 GeV