Neutrons have a rest mass of 1.6749*10-27 kg (equivalent to 939.6 MeV).
If a certain neutron has a total energy of 949 MeV, what are its relativistic mass energy in MeV, its speed in m/s, its kinetic energy in MeV, and its momentum in kg.m.s-1?
m₀=1.6749•10⁻²⁷kg
E=949 MeV=1.52•10⁻¹⁰ J
E=mc² =
m=E/c² =1.52•10⁻¹⁰/(3•10⁸)²=1.6889•10⁻²⁷kg
m=m₀/√(1- β²)
(1- β²)=(m₀/m)²= (1.6749•10⁻²⁷/1.6889•10⁻²⁷)²=0.983
β =sqrt(1-0.983)=0.128
v= β•c=0.128•3•10⁸=3.85•10⁷m/s
m₀c²=1.6749•10⁻²⁷•(3•10⁸)²=1.5074•10⁻¹⁰ J =940.8 MeV
KE=mc²- m₀c²= 949-940.8 =8.2 Mev
p=m₀ βc /√(1- β²)= ….
i caught you
To calculate the relativistic mass energy of the neutron, we can use Einstein's mass-energy equivalence formula:
E = mc^2, where E is the total energy, m is the relativistic mass, and c is the speed of light.
Given that the total energy E is 949 MeV, we can rearrange the formula to solve for m:
m = E / c^2
We know the speed of light, c, is approximately 3.0 x 10^8 m/s. Plugging in the values:
m = 949 MeV / (3.0 x 10^8 m/s)^2
Now we can calculate m:
m ≈ 1.055 x 10^-10 kg
To find the speed of the neutron, we can use the relativistic energy-momentum equation:
E^2 = (mc^2)^2 + (pc)^2, where p is the momentum.
Since we know the total energy E and the relativistic mass m, we can solve for p:
p = sqrt(E^2 - (mc^2)^2) / c
Plugging in the values:
p ≈ sqrt((949 MeV)^2 - (1.055 x 10^-10 kg * (3.0 x 10^8 m/s)^2)^2) / 3.0 x 10^8 m/s
Now we can calculate p:
p ≈ 574.65 kg.m/s
To find the kinetic energy of the neutron, we subtract the rest energy (939.6 MeV) from the total energy (949 MeV):
Kinetic energy = Total energy - Rest energy
Kinetic energy = 949 MeV - 939.6 MeV
Now we can calculate the kinetic energy:
Kinetic energy ≈ 9.4 MeV
Therefore, for a neutron with a total energy of 949 MeV:
- Its relativistic mass energy is approximately 1.055 x 10^-10 kg
- Its speed is approximately 3.0 x 10^8 m/s (approximately the speed of light)
- Its kinetic energy is approximately 9.4 MeV
- Its momentum is approximately 574.65 kg.m/s
To find the relativistic mass energy of the neutron, we can use the equation for the relativistic energy:
E^2 = (mc^2)^2 + (pc)^2
Where:
E is the total energy of the neutron
m is the rest mass of the neutron
c is the speed of light (approximately 3 x 10^8 m/s)
p is the momentum of the neutron
Given that the total energy of the neutron is 949 MeV, we can plug in the values into the equation:
(949 MeV)^2 = (mc^2)^2 + (pc)^2
(949 MeV)^2 - (mc^2)^2 = (pc)^2
Now, we know that the rest mass of the neutron is 939.6 MeV, so we can substitute it into the equation:
(949 MeV)^2 - (939.6 MeV)^2 = (pc)^2
Solving for (pc)^2:
(949 MeV)^2 - (939.6 MeV)^2 = (pc)^2
(949^2 - 939.6^2) MeV^2 = (pc)^2
(949^2 - 939.6^2) MeV^2 = p^2c^2
Dividing both sides by c^2:
[(949^2 - 939.6^2) MeV^2] / c^2 = p^2
Now, we can find the momentum of the neutron by taking the square root of both sides:
p = sqrt([(949^2 - 939.6^2) MeV^2] / c^2)
Calculating this expression will give us the numerical value for the momentum of the neutron in kg.m/s.
To find the speed of the neutron in m/s, we can use the equation for momentum:
p = mv
Where:
p is the momentum of the neutron
m is the relativistic mass of the neutron
v is the velocity of the neutron
Since we now know the momentum of the neutron from the previous calculation, we can rearrange the equation:
v = p / m
Now, we can calculate the speed of the neutron by dividing the momentum (in kg.m/s) by the relativistic mass (in kg).
Finally, to find the kinetic energy of the neutron, we can subtract the rest energy from the total energy of the neutron:
Kinetic energy = Total energy - Rest energy
Now that we have obtained the formulas and explained the steps to find the answers, we can use a calculator to perform the actual calculations based on the given values of rest mass and total energy of the neutron.