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A particle $P$ is formed due to a completely inelastic collision of particles $x$ and $y$ having de-Broglie wavelengths $λ_{x}$ and $λ_{y}$ respectively. If $x$ and $y$ were moving in opposite directions, then the de-Broglie wavelength of $P$ is

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$λ_{x} - λ_{y}$

$\frac{λ_{x} λ_{y}}{|λ_{x} - λ_{y}|}$

$\frac{λ_{x} λ_{y}}{λ_{x} + λ_{y}}$

$λ_{x} + λ_{y}$

answer is A.

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Detailed Solution

$p_{x} = \frac{h}{λ_{x}} and p_{y} = \frac{h}{λ_{y}}$

Since particles are moving in opposite direction, magnitude of resultant momentum of P

$p = |p_{x} - p_{y}| = h |\frac{1}{λ_{x}} - \frac{1}{λ_{y}}|$

$\Rightarrow \frac{h}{λ} = h |\frac{1}{λ_{x}} - \frac{1}{λ_{y}}| \Rightarrow \frac{1}{λ} = \frac{|λ_{y} - λ_{x}|}{λ_{x} λ_{y}} \Rightarrow λ = \frac{λ_{x} λ_{y}}{|λ_{x} - λ_{y}|}$

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