A nucleus of uranium decays at rest into nuclei of thorium and helium. Then :

Options

A. 1. the helium nucleus has more kinetic energy than the thorium nucleus
B. 2. the helium nucleus has less momentum than the thorium nucleus
C. 3. the helium nucleus has more momentum than the thorium nucleus
D. 4. the helium nucleus has less kinetic energy than the thorium nucleus

Answer

To solve the problem of a uranium nucleus decaying at rest into thorium and helium nuclei, we will follow these steps:

### Step 1: Understand the conservation of momentum
When a nucleus decays, the total momentum before and after the decay must be conserved. Since the uranium nucleus is initially at rest, its initial momentum is zero. Therefore, the momentum of the thorium nucleus and the helium nucleus after the decay must also sum to zero.

### Step 2: Set up the momentum equation
Let \( p_{Th} \) be the momentum of the thorium nucleus and \( p_{He} \) be the momentum of the helium nucleus. According to the conservation of momentum:
\[ 0 = p_{Th} + p_{He} \]
This implies:
\[ p_{Th} = -p_{He} \]
This means that the magnitudes of the momenta are equal:
\[ |p_{Th}| = |p_{He}| \]

### Step 3: Relate momentum to kinetic energy
The kinetic energy (KE) of an object can be expressed in terms of its momentum:
\[ KE = \frac{p^2}{2m} \]
For the helium nucleus:
\[ KE_{He} = \frac{p_{He}^2}{2m_{He}} \]
For the thorium nucleus:
\[ KE_{Th} = \frac{p_{Th}^2}{2m_{Th}} \]

### Step 4: Substitute the momentum relationship
Since \( |p_{Th}| = |p_{He}| \), we can substitute \( p_{Th} \) with \( p_{He} \) in the kinetic energy equations:
\[ KE_{Th} = \frac{p_{He}^2}{2m_{Th}} \]
\[ KE_{He} = \frac{p_{He}^2}{2m_{He}} \]

### Step 5: Compare the kinetic energies
Now we can compare the two kinetic energies. Since both expressions have the same numerator \( p_{He}^2 \), the difference in kinetic energy will depend on the masses:
- \( m_{Th} \) (mass of thorium) is greater than \( m_{He} \) (mass of helium).

This means:
\[ KE_{He} > KE_{Th} \]
because the denominator for \( KE_{He} \) is smaller than that for \( KE_{Th} \).

### Conclusion
Thus, we conclude that the helium nucleus has more kinetic energy than the thorium nucleus.

### Final Answer
The helium nucleus has more kinetic energy than the thorium nucleus.

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