A nucleus of mass M initially at rest splits into two fragments of masses `M/3 and (2M)/3`. Find the ratio of de-Broglie wavelength of the fragments.

Explanation

To solve the problem of finding the ratio of de-Broglie wavelengths of the two fragments resulting from the split of a nucleus, we can follow these steps:

### Step 1: Understand the Problem
We have a nucleus of mass \( M \) that is initially at rest. It splits into two fragments with masses \( \frac{M}{3} \) and \( \frac{2M}{3} \). We need to find the ratio of their de-Broglie wavelengths.

### Step 2: Use the Conservation of Momentum
Since the nucleus is initially at rest, the total initial momentum is zero. According to the conservation of momentum, the total momentum after the split must also be zero. Therefore, we can write:

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