In: Physics

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

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

E = m c^{2}

m = m_{o} / SQRT(1 - v^{2}/c^{2})

E_{o} = m_{o} c^{2}

KE = E - E_{o}

KE = m_{o} c^{2} [ {1/SQRT(1 -
v^{2}/c^{2})} - 1]

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

E_{o} = (1.672 x 10 ^{- 27} kg)(3 x
10^{8} 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
10^{9}.

E = m c^{2}

m = m_{o} / SQRT(1 - v^{2}/c^{2})

E_{orig} = [m_{o} / SQRT(1 -
v_{orig}^{2}/c^{2})] c^{2}

E_{orig} = [m_{o} c^{2}][1/ SQRT(1 -
v_{orig}^{2}/c^{2})]

E_{orig} = [313.5 Gev][1/ SQRT(1 - {0.50
c}^{2}/c^{2})]

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 -
va^{2}/c^{2})] c^{2}

E_{a} = [m_{o} c^{2}][1/ SQRT(1 -
v_{a}^{2}/c^{2})]

E_{a} = [313.5 Gev][1/ SQRT(1 - {0.750
c}^{2}/c^{2})]

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_{b}^{2}/c^{2})] c^{2}

E_{b} = [m_{o} c^{2}][1/ SQRT(1 -
v_{b}^{2}/c^{2})]

E_{b} = [313.5 Gev][1/ SQRT(1 - {0.995
c}^{2}/c^{2})]

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**

A proton in a high-energy accelerator moves with a speed of
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