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

Explanation

To solve the problem of finding the recoil speed of the residual nucleus after a Uranium-238 nucleus decays by 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. Therefore, its initial momentum is zero.

### Step 2: Identify the Products of the Decay
When Uranium-238 decays, it emits an alpha particle (which is a Helium-4 nucleus) and leaves behind a residual nucleus. The mass number of the residual nucleus can be calculated as follows:
- Mass number of Uranium-238 = 238
- Mass number of alpha particle = 4
- Mass number of residual nucleus = 238 - 4 = 234

### Step 3: Set Up the Conservation of Momentum Equation
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]

Since the initial momentum is zero, we can write:
\[ 0 = (m_{\alpha} \cdot u) + (m_{residual} \cdot v) \]

Where:
- \( m_{\alpha} \) is the mass of the alpha particle (which corresponds to its mass number, 4).
- \( u \) is the speed of the emitted alpha particle.
- \( m_{residual} \) is the mass of the residual nucleus (which corresponds to its mass number, 234).
- \( v \) is the recoil speed of the residual nucleus.

### Step 4: Substitute the Masses and Rearrange
Substituting the mass numbers for the respective particles:
\[ 0 = (4 \cdot u) + (234 \cdot v) \]

Rearranging the equation gives:
\[ 234 \cdot v = -4 \cdot u \]
\[ v = -\frac{4}{234} \cdot u \]

### Step 5: Simplify the Expression
Now, simplifying the fraction:
\[ v = -\frac{2}{117} \cdot u \]

### Conclusion
The recoil speed of the residual nucleus is:
\[ v = -\frac{2u}{117} \]

The negative sign indicates that the direction of the recoil speed is opposite to that of the emitted alpha particle.