Two particles of de-broglie wavelength `lamda_(x)` and `lamda_(y)` are moving in opposite direction. Find debroglie wavelength after perfectly inelastic collision:

Options

A. `(lamda_(x)lamda_(y))/(|lamda_(x)-lamda_(y)|)`
B. `(2lamda_(x)lamda_(y))/(lamda_(x)-lamda_(y))`
C. `(lamda_(x)lamda_(y)^(2))/(lamda_(y)-lamda_(x))`
D. `lamda_(y)-lamda_(x)`

Answer

To find the de Broglie wavelength after a perfectly inelastic collision between two particles moving in opposite directions, we can follow these steps:

### Step 1: Understand the de Broglie wavelength and momentum
The de Broglie wavelength (\( \lambda \)) of a particle is related to its momentum (\( p \)) by the formula:
\[ p = \frac{h}{\lambda} \]
where \( h \) is Planck's constant.

### Step 2: Define the momenta of the two particles
Let the two particles have de Broglie wavelengths \( \lambda_x \) and \( \lambda_y \). The momenta of the two particles can be expressed as:
\[ p_x = \frac{h}{\lambda_x} \quad \text{(for particle with wavelength } \lambda_x\text{)} \]
\[ p_y = \frac{h}{\lambda_y} \quad \text{(for particle with wavelength } \lambda_y\text{)} \]
Since the particles are moving in opposite directions, we can assign the momentum of the first particle as positive and the second particle as negative:
\[ p_x + (-p_y) = \frac{h}{\lambda_x} - \frac{h}{\lambda_y} \]

### Step 3: Apply conservation of momentum
In a perfectly inelastic collision, the two particles stick together and move with a common momentum after the collision. Let \( \lambda \) be the de Broglie wavelength of the combined particle after the collision. The total momentum after the collision can be expressed as:
\[ p_{\text{total}} = \frac{h}{\lambda} \]
By conservation of momentum, we have:
\[ \frac{h}{\lambda_x} - \frac{h}{\lambda_y} = \frac{h}{\lambda} \]

### Step 4: Simplify the equation
We can factor out \( h \) from the equation:
\[ \frac{1}{\lambda_x} - \frac{1}{\lambda_y} = \frac{1}{\lambda} \]
This can be rewritten as:
\[ \frac{1}{\lambda} = \frac{\lambda_y - \lambda_x}{\lambda_x \lambda_y} \]

### Step 5: Solve for \( \lambda \)
Taking the reciprocal of both sides gives us:
\[ \lambda = \frac{\lambda_x \lambda_y}{\lambda_y - \lambda_x} \]
To ensure that the denominator is positive, we can express it as:
\[ \lambda = \frac{\lambda_x \lambda_y}{|\lambda_y - \lambda_x|} \]

### Final Answer
Thus, the de Broglie wavelength after the perfectly inelastic collision is:
\[ \lambda = \frac{\lambda_x \lambda_y}{|\lambda_y - \lambda_x|} \]

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