Question

Protons in cosmic rays have been observed with kinetic energies as large as 1.0×1020eV.

How fast are these protons moving? Give your answer as a fraction of the speed of light. - the answer is not 4.6*10^5c I've already tried that.

Show that the kinetic energy of a single one of these protons is much greater than the kinetic energy of a 15 mg ant walking with a speed of 8.8 mm/s.

Answer 1

here,

let the speed of protons be v

kinetic energy , KE = 1 * 10^20 eV

m0 * c^2 /(1/sqrt(1 - (v/c)^2) - 1) = 1 * 10^20 * 1.6 * 10^-19

1.67 * 10^-27 * (3 * 10^8)^2 /(1/sqrt(1 - (v/(3 * 10^8))^2) - 1) = 1 * 10^20 * 1.6 * 10^-19

solving for v

v = 3 * 10^8 m/s = c

mass of ant , m' = 15 mg = 0.015 g

speed of ant , v' = 8.8 mm/s = 0.0088 m/s

the kinetic energy for ant , KE' = 0.5 * m' * v'^2

KE' = 0.5 * 0.015 * 0.0088^2 = 5.8 * 10^-7 J

kinetic energy of proton , KEp = 1 * 10^20 eV = 1 * 10^20 * 1.6 * 10^-19 = 16 J

so, the kinetic energy of each proton is much greater than ant