A `U^(238)` nucleus , initially at rests , emits an alpha particle with a speed `v_(0)`. Calculate the recoil speed of the residual nucleus `Th^(234)`. Assume that the mass of a nucleus is proportional to the mass number. Also , calculate the ratio of `K.E. of Th^(234)` and `alpha` -particle.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a Uranium-238 nucleus that is initially at rest and emits an alpha particle. We need to find the recoil speed of the residual Thorium-234 nucleus and the ratio of their kinetic energies.

### Step 2: Define the Masses
- Mass of Uranium-238 nucleus \( (U^{238}) \) = \( 238M \)
- Mass of alpha particle \( (He^{4}) \) = \( 4M \)
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