In: Physics

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

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

## Solutions

##### Expert Solution

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 = mo / SQRT(1 - v2/c2)

Eo = mo c2

KE = E - Eo

KE = mo 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.

Eo = mo c2

Eo = (1.672 x 10 - 27 kg)(3 x 108 m/s)2

Eo = 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,

Eo = 5.016 x 10 - 11 J [1 MeV / 1.6 x 10 - 16 J]

Eo = 313 500 Mev

Eo = 313.5 Gev

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

E = m c2

m = mo / SQRT(1 - v2/c2)

Eorig = [mo / SQRT(1 - vorig2/c2)] c2

Eorig = [mo c2][1/ SQRT(1 - vorig2/c2)]

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

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

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

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

Eorig = [313.5 Gev][1/ 0.866]

Eorig = [313.5 Gev][1.155]

Eorig = 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)?

Ea = [mo / SQRT(1 - va2/c2)] c2

Ea = [mo c2][1/ SQRT(1 - va2/c2)]

Ea = [313.5 Gev][1/ SQRT(1 - {0.750 c}2/c2)]

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

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

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

Ea = 313.5 Gev][1/ 0.661]

Ea = [313.5 Gev][1.512}

Ea = 474 GeV

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

Ea = Ea - Eorig

Ea = (474 - 362) GeV

Ea = 112 GeV

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

Eb = [mo / SQRT(1 - vb2/c2)] c2

Eb = [mo c2][1/ SQRT(1 - vb2/c2)]

Eb = [313.5 Gev][1/ SQRT(1 - {0.995 c}2/c2)]

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

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

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

Eb = [313.5 Gev][1/ 0.09987]

Eb = [313.5 Gev][10.012]

Eb = 3 139 GeV

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

Eb = Eb - Eorig

Eb = (3 139 - 362) GeV

Eb = 2 777 GeV

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