A stationary particle explodes into two particles of masses x and y, which move in opposite directions wit h velocity `v_(1)and v_(2).` The ratio of their kinetic energies `(E_(1):E_(2))` is

Options

A. 1
B. `(xv_(2))/(yv_(1))`
C. `x/y`
D. `y/x`

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 \( x \) and \( y \), moving in opposite directions with velocities \( v_1 \) and \( v_2 \). We need to find the ratio of their kinetic energies \( E_1 : E_2 \).

### Step 2: Apply the conservation of momentum
Since the initial momentum of the stationary particle is zero, by the law of conservation of momentum, the total momentum after the explosion must also be zero. This gives us the equation:

\[ x v_1 - y v_2 = 0 \]

### Step 3: Rearranging the momentum equation
From the momentum equation, we can express the relationship between the velocities:

\[ x v_1 = y v_2 \]

Dividing both sides by \( y v_2 \):

\[ \frac{v_1}{v_2} = \frac{y}{x} \]

### Step 4: Write the expressions for kinetic energies
The kinetic energy of each particle is given by the formula:

\[ E_1 = \frac{1}{2} x v_1^2 \]
\[ E_2 = \frac{1}{2} y v_2^2 \]

### Step 5: Find the ratio of kinetic energies
Now, we can find the ratio of the kinetic energies \( \frac{E_1}{E_2} \):

\[ \frac{E_1}{E_2} = \frac{\frac{1}{2} x v_1^2}{\frac{1}{2} y v_2^2} \]

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

\[ \frac{E_1}{E_2} = \frac{x v_1^2}{y v_2^2} \]

### Step 6: Substitute the ratio of velocities
Substituting \( \frac{v_1}{v_2} = \frac{y}{x} \) into the ratio of kinetic energies:

\[ \frac{E_1}{E_2} = \frac{x \left( \frac{y}{x} v_2 \right)^2}{y v_2^2} \]

This simplifies to:

\[ \frac{E_1}{E_2} = \frac{x \cdot \frac{y^2}{x^2} v_2^2}{y v_2^2} \]

### Step 7: Simplify the expression
Canceling \( v_2^2 \) and simplifying further:

\[ \frac{E_1}{E_2} = \frac{y^2}{x^2} \cdot \frac{x}{y} = \frac{y}{x} \]

### Final Result
Thus, the ratio of the kinetic energies \( E_1 : E_2 \) is:

\[ E_1 : E_2 = y : x \]