In a particular fission process of atom at rest Fragents with mass number `A_(1)` and `A_(2)` are produced. Then find ratio of kinetic energies of the fragments?

Options

A. `A_(2)//A_(1)`
B. `(A_(2)//A_(1))^(2)`
C. `A_(1)//A_(2)`
D. `(A_(1)//A_(2))^(2)`

Answer

To find the ratio of kinetic energies of the fragments produced in a fission process, we can follow these steps:

### Step 1: Understand the Conservation of Momentum
In a fission process, the total momentum before and after the fission must be conserved. Since the original atom is at rest, its initial momentum is zero. Therefore, the momentum of the two fragments after fission must also sum to zero.

### Step 2: Set Up the Momentum Equation
Let the mass numbers of the two fragments be \( A_1 \) and \( A_2 \), and their velocities be \( V_1 \) and \( V_2 \) respectively. According to the conservation of momentum:
\[ A_1 V_1 + A_2 V_2 = 0 \]
This implies:
\[ A_1 V_1 = -A_2 V_2 \]
Taking the magnitudes, we have:
\[ A_1 V_1 = A_2 V_2 \]
From this, we can express the ratio of the velocities:
\[ \frac{V_1}{V_2} = \frac{A_2}{A_1} \]

### Step 3: Write the Kinetic Energy Expressions
The kinetic energy (KE) of each fragment can be expressed as:
\[ KE_1 = \frac{1}{2} A_1 V_1^2 \]
\[ KE_2 = \frac{1}{2} A_2 V_2^2 \]

### Step 4: Find the Ratio of Kinetic Energies
Now, we can find the ratio of the kinetic energies:
\[ \frac{KE_1}{KE_2} = \frac{\frac{1}{2} A_1 V_1^2}{\frac{1}{2} A_2 V_2^2} = \frac{A_1 V_1^2}{A_2 V_2^2} \]
The \( \frac{1}{2} \) cancels out, so we simplify to:
\[ \frac{KE_1}{KE_2} = \frac{A_1 V_1^2}{A_2 V_2^2} \]

### Step 5: Substitute the Velocity Ratio
Using the ratio \( \frac{V_1}{V_2} = \frac{A_2}{A_1} \), we can substitute \( V_1 \) in terms of \( V_2 \):
\[ V_1 = \frac{A_2}{A_1} V_2 \]
Now substituting this into the kinetic energy ratio:
\[ \frac{KE_1}{KE_2} = \frac{A_1 \left( \frac{A_2}{A_1} V_2 \right)^2}{A_2 V_2^2} \]
This simplifies to:
\[ \frac{KE_1}{KE_2} = \frac{A_1 \frac{A_2^2}{A_1^2} V_2^2}{A_2 V_2^2} = \frac{A_2}{A_1} \]

### Final Answer
Thus, the ratio of the kinetic energies of the fragments is:
\[ \frac{KE_1}{KE_2} = \frac{A_2}{A_1} \]