What is the kinetic energy in MeV of a π–meson that lives 1.80×10-16 s as measured in the laboratory, and 0.840×10-16 s when at rest relative to an observer, given that its rest energy is 135 MeV?
To calculate the kinetic energy of a particle, we need to use the equation:
Kinetic energy (KE) = Total energy - Rest energy
Total energy of a particle can be calculated using Einstein's mass-energy equivalence equation:
Total energy = Rest energy + Kinetic energy
Given that the rest energy of the π–meson is 135 MeV, let's calculate its total energy when it is at rest relative to an observer:
Total energy (rest) = Rest energy = 135 MeV
Now, let's calculate the total energy of the π–meson when it is measured in the laboratory:
Total energy (lab) = Rest energy + Kinetic energy (lab)
We can rearrange the equation and solve for the kinetic energy (lab):
Kinetic energy (lab) = Total energy (lab) - Rest energy
To calculate the total energy (lab), we need to account for time dilation. Time dilation is the difference in the passage of time between two different observers moving relative to each other. The time dilation equation is:
Δt = γ * Δt0
where Δt is the measured time in the laboratory, Δt0 is the proper time (time when the particle is at rest), and γ is the Lorentz factor given by:
γ = 1 / √(1 - (v^2 / c^2))
where v is the velocity of the π–meson and c is the speed of light (3.00 x 10^8 m/s).
Let's start by calculating the Lorentz factor (γ) using the observed time (Δt) and proper time (Δt0):
γ = Δt / Δt0
γ = (1.80 x 10^(-16) s) / (0.840 x 10^(-16) s)
Now, let's substitute the Lorentz factor (γ) into the equation for total energy (lab):
Total energy (lab) = γ * Rest energy
Now we can calculate the total energy (lab):
Total energy (lab) = γ * 135 MeV
Finally, let's substitute the values of Total energy (lab) and Rest energy into the equation for kinetic energy (lab):
Kinetic energy (lab) = Total energy (lab) - Rest energy
By doing these calculations, we can determine the kinetic energy of the π–meson in MeV when measured in the laboratory.