A stationary particle explodes into two particle of a masses `m_(1) and m_(2)` which move in opposite direction with velocities `v_(1) and v_(2)`. The ratio of their kinetic energies `E_(1)//E_(2)` is

Options

A. 1
B. `(m_(1)v_(1))/(m_(2)v_(2))`
C. `(m_(2))/(m_(1))`
D. `(m_(1))/(m_(2))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a stationary particle that explodes into two particles with masses \( m_1 \) and \( m_2 \) moving in opposite directions with velocities \( v_1 \) and \( v_2 \). We need to find the ratio of their kinetic energies \( \frac{E_1}{E_2} \).

### Step 2: Apply Conservation of Momentum
Since the initial momentum of the stationary particle is zero, the momentum after the explosion must also be zero. Therefore, we can write:

\[ m_1 v_1 + m_2 (-v_2) = 0 \]

This simplifies to:

\[ m_1 v_1 = m_2 v_2 \]

### Step 3: Write the Kinetic Energy Expressions
The kinetic energy \( E \) of a particle is given by the formula:

\[ E = \frac{1}{2} m v^2 \]

Thus, the kinetic energies of the two particles are:

\[ E_1 = \frac{1}{2} m_1 v_1^2 \]
\[ E_2 = \frac{1}{2} m_2 v_2^2 \]

### Step 4: Find the Ratio of Kinetic Energies
Now, we can find the ratio of their kinetic energies:

\[ \frac{E_1}{E_2} = \frac{\frac{1}{2} m_1 v_1^2}{\frac{1}{2} m_2 v_2^2} \]

The \( \frac{1}{2} \) cancels out:

\[ \frac{E_1}{E_2} = \frac{m_1 v_1^2}{m_2 v_2^2} \]

### Step 5: Substitute Using Momentum Relation
From our momentum relation \( m_1 v_1 = m_2 v_2 \), we can express \( v_2 \) in terms of \( v_1 \):

\[ v_2 = \frac{m_1}{m_2} v_1 \]

Now substitute \( v_2 \) into the ratio:

\[ \frac{E_1}{E_2} = \frac{m_1 v_1^2}{m_2 \left(\frac{m_1}{m_2} v_1\right)^2} \]

This simplifies to:

\[ \frac{E_1}{E_2} = \frac{m_1 v_1^2}{m_2 \frac{m_1^2}{m_2^2} v_1^2} \]

Cancelling \( v_1^2 \) and simplifying gives:

\[ \frac{E_1}{E_2} = \frac{m_1}{\frac{m_1^2}{m_2}} = \frac{m_1 m_2}{m_1^2} = \frac{m_2}{m_1} \]

### Final Result
Thus, the ratio of their kinetic energies is:

\[ \frac{E_1}{E_2} = \frac{m_2}{m_1} \]