Question

A proton has a total energy of E = 3.5 GeV. Determine the following physical quantities: a) speed... v (in terms of c) b) momentum... p ( in units... kg m/s & MeV/c ) c) kinetic energy... K ( in units... J & MeV ).

Answer 1

a)

E = 3.5 GeV = 3.5 x 109 x 1.6 x 10-19 J = 5.6 x 10-10 J

v = speed

m = mass of proton = 1.67 x 10-27 kg

we know that

E = mc2/sqrt(1 - (v/c)2)

5.6 x 10-10 = (1.67 x 10-27)(3 x 108)2/sqrt(1 - (v/(3 x 108))2)

v = 2.88993 x 108 m/s

v = (2.88993 x 108) (c/(3 x 108))

v = 0.963 c

b)

Momentum is given as

p = mv/sqrt(1 - (v/c)2)

p = (1.67 x 10-27) (2.88993 x 108)/sqrt(1 - (0.963c/c)2)

p = 1.8 x 10-18 kgm/s

we know that : 1 MeV/c = 5.36 x 10-22 kgm/s

p = 1.8 x 10-18 kgm/s (1 MeV/c /(5.36 x 10-22 kgm/s ))

p = (1.8 x 10-18/(5.36 x 10-22)) MeV/c

p = 3358.21 MeV/c

c)

kinetic energy is given as

KE = mc2 (1/sqrt(1 - (v/c)2) - 1)

KE = (1.67 x 10-27)(3 x 108)2(1/sqrt(1 - ((2.88993 x 108)/(3 x 108))2 - 1)

KE = 4.10 x 10-10 J

KE = 4.10 x 10-10 J (1 MeV/(1.609 x 10-13) J)

KE = 2548.2 MeV