A normal stationary lithium atom releases an `alpha`-particle with velocity `(2hati+3hatj+7hatk)`km/s .
What will be the velocity of the daughter element just after the releases of the `alpha`-particle ?

Options

A. `(1//2)(2hati+3hatj+7hatk)`km/s
B. `(-4//3)(2hati+3hatj+7hatk)`km/s
C. `(-1//4) (2hati+3hatj+7hatk)` km/s
D. `(1//5)(2hati+3hatj+7hatk)` km/s

Answer

To solve the problem of finding the velocity of the daughter element after the lithium atom releases an alpha particle, we can use the principle of conservation of momentum. Here's a step-by-step solution:

### Step 1: Understand the System
The lithium atom is initially stationary, meaning its initial momentum is zero. When it releases an alpha particle, the momentum of the system must be conserved.

### Step 2: Define the Variables
Let:
- \( m_\alpha \) = mass of the alpha particle (approximately 4 u)
- \( \vec{v}_\alpha \) = velocity of the alpha particle = \( (2 \hat{i} + 3 \hat{j} + 7 \hat{k}) \) km/s
- \( m_d \) = mass of the daughter element (tritium, approximately 3 u)
- \( \vec{v}_d \) = velocity of the daughter element (unknown, to be determined)

### Step 3: Apply Conservation of Momentum
The total momentum before the release of the alpha particle is zero (since the lithium atom is stationary). After the release, the momentum of the alpha particle and the daughter element must add up to zero:

\[ \text{Initial Momentum} = \text{Final Momentum} \]

This gives us the equation:

\[ 0 = m_\alpha \vec{v}_\alpha + m_d \vec{v}_d \]

### Step 4: Rearrange the Equation
Rearranging the equation to solve for the velocity of the daughter element:

\[ m_d \vec{v}_d = -m_\alpha \vec{v}_\alpha \]

### Step 5: Substitute the Known Values
Substituting the masses and the velocity of the alpha particle into the equation:

\[ 3 \vec{v}_d = -4 (2 \hat{i} + 3 \hat{j} + 7 \hat{k}) \]

### Step 6: Solve for \(\vec{v}_d\)
Now, we can solve for \(\vec{v}_d\):

\[ \vec{v}_d = -\frac{4}{3} (2 \hat{i} + 3 \hat{j} + 7 \hat{k}) \]

Calculating the components:

\[ \vec{v}_d = -\frac{8}{3} \hat{i} - 4 \hat{j} - \frac{28}{3} \hat{k} \]

### Step 7: Final Result
Thus, the velocity of the daughter element (tritium) just after the release of the alpha particle is:

\[ \vec{v}_d = -\frac{8}{3} \hat{i} - 4 \hat{j} - \frac{28}{3} \hat{k} \text{ km/s} \]