which a `U^(238)` nucleus original at rest , decay by emitting an alpha particle having a speed `u` , the recoil speed of the residual nucleus is

Options

A. `(4v)/(238)`
B. `-(4v)/(238)`
C. `v/4`
D. `-(4v)/(234)`

Answer

To solve the problem of finding the recoil speed of the residual nucleus after the decay of a Uranium-238 nucleus emitting an alpha particle, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
- The Uranium-238 nucleus is initially at rest, which means its initial velocity \( V_i = 0 \).
- The mass of the Uranium-238 nucleus is \( m_{U} = 238 \) units (in atomic mass units, amu).
- An alpha particle is emitted with a mass \( m_{\alpha} = 4 \) amu and speed \( u \).

### Step 2: Write the Conservation of Momentum Equation
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
Since the Uranium nucleus is at rest initially, the initial momentum is:
\[ 0 = m_{U} \cdot V_i \]
After the decay, the final momentum consists of the momentum of the emitted alpha particle and the recoil momentum of the residual nucleus (Uranium-234):
\[ 0 = m_{\alpha} \cdot u + m_{residual} \cdot V \]
Where:
- \( m_{residual} = 238 - 4 = 234 \) amu (mass of the residual nucleus)
- \( V \) is the recoil speed of the residual nucleus.

### Step 3: Substitute Known Values
Substituting the known values into the momentum equation:
\[ 0 = (4 \cdot u) + (234 \cdot V) \]
Rearranging gives:
\[ 234 \cdot V = -4 \cdot u \]

### Step 4: Solve for the Recoil Speed \( V \)
To find \( V \), we divide both sides by 234:
\[ V = -\frac{4u}{234} \]

### Step 5: Simplify the Expression
This can be simplified further:
\[ V = -\frac{2u}{117} \]

### Conclusion
The recoil speed of the residual nucleus after the alpha decay is:
\[ V = -\frac{4u}{234} \]
The negative sign indicates that the recoil speed is in the opposite direction to the emitted alpha particle.

### Final Answer
The recoil speed of the residual nucleus is:
\[ V = -\frac{4u}{234} \]