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Chapter 6: Problem 152

Find the velocity and kinetic energy of a 6.0 -fm neutron. (Rest mass energy of neutron is \(E_{0}=940 \mathrm{MeV}\).)

Short Answer

Expert verified
The velocity of the 6.0-fm neutron is \(1.724 \times 10^7\) m/s, and its kinetic energy is \(4.1512 \times 10^{-11}\) J.

Step by step solution

01

Convert the rest mass energy to joules

First, we need to convert the given rest mass energy in MeV to joules. 1 electron volt (eV) is equal to 1.602 x 10^(-19) joules. Therefore, we can determine the rest mass energy in joules. E₀ = (940 MeV)(1.602 x 10^(-13) J/MeV) = 1.50588 x 10^(-10) J
02

Find the total energy (E) using energy-mass relation

Next, we need to find the total energy (E) using the energy-mass relation. The formula for the energy-mass relation is: E² = (m₀ c²)² + (p c)² where E is the total energy, m₀ is the rest mass, c is the speed of light, and p is the momentum of the particle. Since we already know the rest mass energy, we can use the following simplification: E² = (E₀)² + (p c)² Since we are working with a neutron, which is a fermion, we can use the Compton wavelength of the neutron, λ = 6.0 fm (1 fm = 10^(-15) m), and the Planck constant, h = 6.626 x 10^(-34) J•s, to find the momentum: p = h / λ Hence, we can find the total energy E.
03

Calculate the momentum of the neutron

Now, let's calculate the momentum of the neutron using the given Compton wavelength. λ = 6.0 x 10^(-15) m p = (6.626 x 10^(-34) J•s) / (6.0 x 10^(-15) m) = 1.104 x 10^(-19) kg•m/s
04

Calculate the total energy E

Using the derived momentum, we can now calculate the total energy E. E² = (1.50588 x 10^(-10) J)² + [(1.104 x 10^(-19) kg•m/s)(3 x 10^8 m/s)]² E = √(2.263 x 10^(-20) J² + 9.829 x 10^(-21) J²) = 1.921 x 10^(-10) J
05

Find the velocity of the neutron

To find the velocity of the neutron, we can use the formula of momentum: p = m v We know that relativistic mass m is related to rest mass m₀ by: m = E / c² Substituting these formulas, we get: v = p c² / E Now, we can find the velocity of the neutron: v = (1.104 x 10^(-19) kg•m/s)(3 x 10^8 m/s)² / (1.921 x 10^(-10) J) = 1.724 x 10^7 m/s
06

Calculate the kinetic energy of the neutron

Finally, we will find the kinetic energy of the neutron using the formula: K.E. = E - E₀ K.E. = (1.921 x 10^(-10) J) - (1.50588 x 10^(-10) J) = 4.1512 x 10^(-11) J So, the velocity of the 6.0-fm neutron is 1.724 x 10^7 m/s, and its kinetic energy is 4.1512 x 10^(-11) J.

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