It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is `p_d,` while for its similar collision with carbon nucleus at rest, fractional loss of energy is `p_c`. The values of `p_d and p_c` are respectively.

Options

A. (0,0)
B. (0,1)
C. (.89, .28)
D. (28, .89)

Explanation

To solve the problem of fractional energy loss during elastic collisions of a neutron with deuterium and carbon, we will follow these steps:

### Step 1: Understand the scenario
We have a neutron colliding elastically with two different nuclei: deuterium (with 2 nucleons) and carbon (with 12 nucleons). We need to find the fractional loss of energy for both collisions.

### Step 2: Use conservation of momentum and kinetic energy
In an elastic collision, both momentum and kinetic energy are conserved. The initial momentum of the neutron is given by:
\[ p_{initial} = mv \]
where \( m \) is the mass of the neutron and \( v \) is its initial velocity.

After the collision, let the velocities of the neutron, deuterium, and carbon be \( v_1 \) and \( v_2 \) respectively.

### Step 3: Set up equations for momentum conservation
For the neutron colliding with a nucleus of mass \( M \) (where \( M \) is the mass of the deuterium or carbon nucleus), we have:
\[ mv = mv_1 + Mv_2 \]

### Step 4: Set up equations for energy conservation
The initial kinetic energy of the neutron is:
\[ KE_{initial} = \frac{1}{2} mv^2 \]
The final kinetic energy after the collision is:
\[ KE_{final} = \frac{1}{2} mv_1^2 + \frac{1}{2} Mv_2^2 \]

### Step 5: Solve for final velocities
Using the equations of conservation of momentum and kinetic energy, we can derive the final velocities. The velocity of the neutron after the collision can be expressed as:
\[ v_1 = \frac{(m - M)v}{(m + M)} \]
where \( M \) is the mass of the nucleus (2 for deuterium and 12 for carbon).

### Step 6: Calculate fractional loss of kinetic energy
The fractional loss of kinetic energy is given by:
\[ \text{Fractional Loss} = \frac{KE_{initial} - KE_{final}}{KE_{initial}} \]

Substituting the expressions for kinetic energy:
\[ \text{Fractional Loss} = 1 - \left(\frac{v_1}{v}\right)^2 \]

### Step 7: Calculate for deuterium
For deuterium (nucleon count \( N = 2 \)):
\[ v_1 = \frac{(1 - 2)v}{(1 + 2)} = \frac{-v}{3} \]
Thus,
\[ \text{Fractional Loss for Deuterium} = 1 - \left(\frac{-1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]

### Step 8: Calculate for carbon
For carbon (nucleon count \( N = 12 \)):
\[ v_1 = \frac{(1 - 12)v}{(1 + 12)} = \frac{-11v}{13} \]
Thus,
\[ \text{Fractional Loss for Carbon} = 1 - \left(\frac{-11}{13}\right)^2 = 1 - \frac{121}{169} = \frac{48}{169} \]

### Final Result
The fractional losses of energy are:
- For deuterium: \( p_d = \frac{8}{9} \)
- For carbon: \( p_c = \frac{48}{169} \)