Question

A proton in a high-energy accelerator moves with a speed of c/2. Use the work

Answer 1

For common, ordinary speeds, we found that the (kinetic) energy of a particle with mass m and speed v is KE = (1/2) m v². But for very high speeds -- for speeds close to that of light -- we had to modify that and use

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]

This does not look very much like KE = (1/2) m v² ! So it is worth reminding ourselves tha this unusual looking expression does, indeed, reduce to KE = (1/2) m v² for v << c !

First, let's find the "rest energy" of the proton since this factor appears in all our future calculations.

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

This is perfectly okay as it is. However, instead of dealing with such small numbers, it is customary to talk about these energies in units of MeV -- millions of electron-Volts. We will explore MeV's more in PHY 1360 (next semester). For the present, we can simply pull out of the air the appropriate conversion factor,

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

One GeV is one giggaelectron-volt. Gigga means "billion" or 10⁹.

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

That's the total energy -- KE plus "rest mass energy" -- that the proton has originally. Now, how much energy does it have for the speeds given in question (a) and question (b)?

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 - {0.750 c}²/c²)]

E_a = [313.5 Gev][1/ SQRT(1 - {0.750}²)]

E_a = [313.5 Gev][1/ SQRT(1 - 0.5625)]

E_a = [313.5 Gev][1/ SQRT(0.4375)]

E_a = 313.5 Gev][1/ 0.661]

E_a = [313.5 Gev][1.512}

E_a = 474 GeV

How much energy do we need to add to go from the original speed (v_orig = 0.500 c) tho this speed (v_a = 0.75 c)?

E_a = E_a - E_orig

E_a = (474 - 362) GeV

E_a = 112 GeV

Now we do exactly the same thing for v_b = 0.995 c.

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 - {0.995 c}²/c²)]

E_b = [313.5 Gev][1/ SQRT(1 - {0.995}²)]

E_b = [313.5 Gev][1/ SQRT(1 - 0.990025)]

E_b = [313.5 Gev][1/ SQRT(0.009975)]

E_b = [313.5 Gev][1/ 0.09987]

E_b = [313.5 Gev][10.012]

E_b = 3 139 GeV

How much energy do we need to add to go from the original speed (v_orig = 0.500 c) tho this speed (v_b = 0.995 c)?

E_b = E_b - E_orig

E_b = (3 139 - 362) GeV

E_b = 2 777 GeV