Question

What is the kinetic energy in MeV of a π-meson that lives 1.40×10^(-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?

a) 60 MeV
b) 75 MeV
c) 90 MeV
d) 120 MeV

Answer 1

The kinetic energy of the π-meson is approximately 90 MeV, calculated by first finding its velocity using time dilation and then using the relativistic kinetic energy formula.

Step-by-step explanation:

To find the kinetic energy of a π-meson, we can start by calculating its velocity using the time measurements given. The change in time between the laboratory measurement and the observer at rest measurement gives us the time dilation factor. Using the formula for time dilation and the rest energy of the π-meson, we can calculate its velocity. With the velocity, we can then calculate the relativistic kinetic energy using the formula for kinetic energy. Converting the kinetic energy from eV to MeV gives us the answer.

First, let's calculate the velocity:

Given: Δt_lab = 1.40 × 10-16 s, Δt_rest = 0.840 × 10-16 s, Erest = 135 MeV

Time dilation factor (γ) = Δt_lab / Δt_rest

γ = 1.40 × 10-16 s / 0.840 × 10-16 s

γ = 1.667

Velocity (v) = c / √(γ2 - 1)

v = c / √(1.6672 - 1)

v ≈ 0.8935c

Now, let's calculate the kinetic energy:

Rest energy (Erest) = 135 MeV

Relativistic kinetic energy (KE) = (γ - 1) * Erest

KE = (1.667 - 1) * 135 MeV

KE ≈ 0.667 * 135 MeV

KE ≈ 90 MeV

So, the kinetic energy of the π-meson is approximately 90 MeV.