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Step-by-Step Solution
Step 1: State the problem
A particle of total mass $4M$, initially at rest, disintegrates into two particles of masses $M$ and $3M$. We need to find the ratio of the de-Broglie wavelength of the smaller mass ($M$) to that of the larger mass ($3M$).
Step 2: Apply conservation of momentum
Because the original particle was at rest, its total momentum was zero. After disintegration, if the momentum of the mass $M$ is $p$ in one direction, then the momentum of the mass $3M$ must be $p$ in the opposite direction to conserve the total momentum at zero.
Step 3: Write the de-Broglie wavelength formula
The de-Broglie wavelength $ \lambda $ for a particle is given by
$ \lambda = \frac{h}{p}, $
where $h$ is Planck's constant and $p$ is the magnitude of the particle’s momentum.
Step 4: Compare the wavelengths
Since both particles have the same magnitude of momentum $p$ (though in opposite directions), their de-Broglie wavelengths are:
$ \lambda_M = \frac{h}{p} $ for the particle of mass $M$,
$ \lambda_{3M} = \frac{h}{p} $ for the particle of mass $3M$.
Thus, the ratio of these wavelengths is:
$ \frac{\lambda_M}{\lambda_{3M}} = \frac{\frac{h}{p}}{\frac{h}{p}} = 1. $
Step 5: State the final ratio
Therefore, the ratio of the de-Broglie wavelength of the particle of mass $M$ to that of mass $3M$ is 1 : 1.
