Final answer:
The speed of the proton is approximately 0.9999997599 c, and its momentum is approximately 4.007 x 10^-19 GeV/c.
Step-by-step explanation:
To find the speed of a proton with a total energy of 4.200 GeV, we can first convert the energy to kinetic energy using the equation: Kinetic energy = total energy - rest energy. The rest energy of a proton is approximately 0.938 GeV, so the kinetic energy is 4.200 GeV - 0.938 GeV = 3.262 GeV. Next, we can use the equation: Kinetic energy = (1/2)mv^2, where m is the mass of the proton and v is its speed. Rearranging the equation, we have: v = sqrt((2 * Kinetic energy) / m). The mass of a proton is approximately 1.67 x 10^-27 kg. Plugging in the values, we get: v = sqrt((2 * 3.262 GeV) / (1.67 x 10^-27 kg)). Calculating this, we find that the speed of the proton is approximately 0.9999997599 c, where c is the speed of light.
To calculate the momentum of the proton in GeV/c, we can use the equation: Momentum = mass * velocity. Plugging in the values, we get: Momentum = (1.67 x 10^-27 kg) * (0.9999997599 c). Calculating this, we find that the momentum of the proton is approximately 4.007 x 10^-19 GeV/c.