In a location in outer space far from all other objects, a nucleus whose mass is 3.57232e-25 kg and which is initially at rest undergoes spontaneous "alpha" decay. The original nucleus disappears, and two new particles appear: a He-4 nucleus of mass 6.640678e-27 kg (an "alpha particle" consisting of two protons and two neutrons) and a new nucleus of mass 3.50582e-25 kg. These new particles move far away from each other, because they repel each other electrically (both are positively charged).
Because the calculations involve the small difference of (comparatively) large numbers, you need to keep 7 significant figures in your calculations, and you need to use the more accurate value for the speed of light, 2.99792e8 m/s.
Choose all particles as the system.
Initial state: Original nucleus, at rest.
Final state: Alpha particle + new nucleus, far from each other.
What is the rest energy of the original nucleus? Give 7 significant figures.
What is the sum of the rest energies of the alpha particle and the new nucleus? Give 7 significant figures.
Rest energy of the original nucleus: Well, it's like that old saying, "Rest in peace"... or in this case, "Rest energy in kilogram times the speed of light squared." So let's do some calculations, shall we?
The rest energy of the original nucleus can be calculated using the famous Einstein's equation E=mc². So, E = (3.57232e-25 kg) * (2.99792e8 m/s)²
Calculating that out, we get: 9.0685e-8 J (rounded to 7 significant figures)
Now onto the sum of the rest energies of the alpha particle and the new nucleus! This one's gonna be a blast (pun intended).
The rest energy of the alpha particle is given as 6.640678e-27 kg, and the rest energy of the new nucleus is given as 3.50582e-25 kg.
Adding those together, we get: (6.640678e-27 kg) + (3.50582e-25 kg) = 3.5690e-25 kg (rounded to 7 significant figures)
So, the sum of the rest energies of the alpha particle and the new nucleus is 3.5690e-25 kg. Voila!
To find the rest energy of the original nucleus, we can use the equation E = mc², where E is the rest energy, m is the mass, and c is the speed of light.
Given:
Mass of the original nucleus = 3.57232e-25 kg
Speed of light (c) = 2.99792e8 m/s
Using the equation, let's calculate the rest energy of the original nucleus:
E = (3.57232e-25 kg) * (2.99792e8 m/s)²
E ≈ 3.206e-8 J
Therefore, the rest energy of the original nucleus is approximately 3.206e-8 J.
To find the sum of the rest energies of the alpha particle and the new nucleus, we can use the same equation, E = mc².
Given:
Mass of the alpha particle = 6.640678e-27 kg
Mass of the new nucleus = 3.50582e-25 kg
Speed of light (c) = 2.99792e8 m/s
Calculating the rest energy of the alpha particle:
E_alpha = (6.640678e-27 kg) * (2.99792e8 m/s)²
E_alpha ≈ 5.929e-11 J
Calculating the rest energy of the new nucleus:
E_nucleus = (3.50582e-25 kg) * (2.99792e8 m/s)²
E_nucleus ≈ 3.145e-8 J
Adding the rest energies of the alpha particle and the new nucleus:
E_total = E_alpha + E_nucleus
E_total ≈ 5.929e-11 J + 3.145e-8 J
E_total ≈ 3.148e-8 J
Therefore, the sum of the rest energies of the alpha particle and the new nucleus is approximately 3.148e-8 J.
To calculate the rest energy of the original nucleus, we can use the formula:
E = mc^2
where E is the rest energy, m is the mass, and c is the speed of light.
Given:
m = 3.57232e-25 kg
c = 2.99792e8 m/s
Substituting these values into the formula, we get:
E = (3.57232e-25 kg) * (2.99792e8 m/s)^2
Calculating this expression, we find:
E ≈ 3.21422e-8 J
So, the rest energy of the original nucleus is approximately 3.21422e-8 Joules.
Now, let's calculate the sum of the rest energies of the alpha particle and the new nucleus. We can use the same formula as before for each particle and then add the results together.
For the alpha particle:
m_alpha = 6.640678e-27 kg
E_alpha = (6.640678e-27 kg) * (2.99792e8 m/s)^2
Calculating this expression, we find:
E_alpha ≈ 5.94953e-11 J
For the new nucleus:
m_new = 3.50582e-25 kg
E_new = (3.50582e-25 kg) * (2.99792e8 m/s)^2
Calculating this expression, we find:
E_new ≈ 3.15019e-8 J
Now, adding the rest energies together:
Total rest energy = E_alpha + E_new
Total rest energy ≈ 5.94953e-11 J + 3.15019e-8 J
Calculating this expression, we find:
Total rest energy ≈ 3.150785e-8 J
Therefore, the sum of the rest energies of the alpha particle and the new nucleus is approximately 3.150785e-8 Joules.
Calculate the loss of rest mass energy. That will equal the sum of the kinetic energies of the two fragments.
(delta m)*c^2 = 9.322*10^-30 kg*c^2
= 8.379*10^-13 J
Since the original particle was stationary, the two fragments have equal and opposite momenta.
You can use that fact and the total final kinetic energy to get the velocities of each of the two fragments. However, all they ask you to do is compute m c^2 for each particle and compare initial and final values. That has already been done above.