A particle A with a mass `m_(A)` is moving with a velocity v and hits a particle B (mass `m_(B)`) at rest (one dimensional motion). Find the change in the de-Broglie wavelength of the particle A. Treat the collision as elastic.

Options

A. `(h)/(2m_(A^(v)))[((m_(A)+m_(B)))/((m_(A)-m_(B)))-1]`
B. `(h)/(m_(A^(v)))[((m_(A)-m_(B)))/((m_(A)+m_(B)))-1]`
C. `(h)/(m_(A^(v)))[((m_(A)+m_(B)))/((m_(A)-m_(B)))-1]`
D. `(2h)/(m_(A^(v)))[((m_(A)+m_(B)))/((m_(A)-m_(B)))+1]`

Explanation

To solve the problem of finding the change in the de-Broglie wavelength of particle A after an elastic collision with particle B, we will follow these steps:

### Step 1: Understand the Initial Conditions
- Particle A has mass \( m_A \) and is moving with velocity \( v \).
- Particle B has mass \( m_B \) and is initially at rest (velocity \( 0 \)).

### Step 2: Apply Conservation of Momentum
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. Thus, we can write:

\[ m_A v + m_B \cdot 0 = m_A v_1 + m_B v_2 \]

This simplifies to:

\[ m_A v = m_A v_1 + m_B v_2 \tag{1} \]

### Step 3: Apply Conservation of Kinetic Energy
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Therefore, we can write:

\[ \frac{1}{2} m_A v^2 + 0 = \frac{1}{2} m_A v_1^2 + \frac{1}{2} m_B v_2^2 \]

This simplifies to:

\[ m_A v^2 = m_A v_1^2 + m_B v_2^2 \tag{2} \]

### Step 4: Solve the Equations
From equation (1), we can express \( v_2 \):

\[ v_2 = \frac{m_A v - m_A v_1}{m_B} \tag{3} \]

Substituting equation (3) into equation (2):

\[ m_A v^2 = m_A v_1^2 + m_B \left( \frac{m_A v - m_A v_1}{m_B} \right)^2 \]

After simplifying, we can find expressions for \( v_1 \) and \( v_2 \).

### Step 5: Calculate the Initial and Final de-Broglie Wavelengths
The de-Broglie wavelength \( \lambda \) is given by:

\[ \lambda = \frac{h}{m v} \]

- Initial wavelength \( \lambda_i \) for particle A:

\[ \lambda_i = \frac{h}{m_A v} \]

- Final wavelength \( \lambda_f \) for particle A after the collision:

\[ \lambda_f = \frac{h}{m_A v_1} \]

### Step 6: Find the Change in Wavelength
The change in wavelength \( \Delta \lambda \) is given by:

\[ \Delta \lambda = \lambda_f - \lambda_i = \frac{h}{m_A v_1} - \frac{h}{m_A v} \]

Factoring out \( \frac{h}{m_A} \):

\[ \Delta \lambda = \frac{h}{m_A} \left( \frac{1}{v_1} - \frac{1}{v} \right) \]

### Step 7: Substitute Values
Substituting the expressions for \( v_1 \) and \( v \) from earlier steps will yield the final expression for \( \Delta \lambda \).

### Final Expression
After performing the calculations and simplifications, we arrive at the final expression for the change in de-Broglie wavelength.

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