Question

Ann measures the (relativistic) momentum of a tiny particle to be 3.2x 103 kg mIs. The mass of the particle is 8 x 10-3g. and it is moving in the -x-direction. SP6 a. Find the total energy of the particle according to Ann b. Find the difference of the particle's total energy and its rest energy, and compare it to the kinetie energy of the particle calculated non-relativistically. Write the components of the momentum 4-vector of the particle according to Ann. Bob moves past Ann in the +x-direction at a speed of 0.6c, and measures the energy of the same particle. Compute the energy of the particle according to Bob in two ways: i. using velocity addition ii. using a lorentz transformation of the 4-vector momentum components measred by Ann. c. d.

Answer 1

Relativistic To total energy of the particle is E = squareroot p^2 c^2 + m^2 c^4 (rho = momentum m = rest mass) = squareroot (3.2 times 10^3)^2 (3 times 10^8)^2 + (8 times 10^-6)^2 times (3 times 10^8)^4 = squareroot 9.22 times 10^23 + 5.18 times 10^23 = 1.2 times 10^12 Required difference between the total energy and rest energy is E_x = squareroot p^2 c^2 + m^2 c^4 - m c^2 = [1.2 times 10^12 - (8 times 10^-6) middot (3 times 10^8)^2] = 4.8 times 10^11 which is the relativistic kinetic energy The non-relativistic kinetic energy is given by E_k non-relativistic = 1/2 m v^2 \r\n velocity middot v can be calculated from P middot m_v /squareroot 1 - v^2 /c^2 = 3.2 times 10^3 or v/squareroot 1 - v^2 /c^2 = 3.2 times 10^3 /8 times 10^-6 = 4 times 10^8 = 4/3 C (3v/4c)^2 = (1 - v^2 /c^2) (9/16) v^2 /c^2 = 1 - v^2 /c^2 25/16 v^2 /c^2