Calculate the uncertainty in velocity if the uncertainty in the position of a moving bullet of mass 10 gm is `10^(-5)m`.

Options

A. `5.2xx10^(-28)m//sec`
B. `3.0xx10^(-28)m//sec`
C. `5.2xx10^(-22)m//sec`
D. `3xx10^(-22)m//sec`

Answer

To calculate the uncertainty in velocity (Δv) using the Heisenberg Uncertainty Principle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data:**
- Mass of the bullet (m) = 10 g = 0.01 kg (since 1 g = 0.001 kg)
- Uncertainty in position (Δx) = \(10^{-5}\) m
- Planck's constant (h) = \(6.626 \times 10^{-34}\) Js

2. **Heisenberg Uncertainty Principle:**
The principle states that:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where Δp is the uncertainty in momentum. Since momentum (p) is given by:
\[ p = mv \]
we can express the uncertainty in momentum as:
\[ \Delta p = m \Delta v \]

3. **Rearranging the Equation:**
From the Heisenberg principle, we can rearrange to find Δv:
\[ \Delta v = \frac{h}{4\pi m \Delta x} \]

4. **Substituting the Values:**
Now, substitute the values into the equation:
\[ \Delta v = \frac{6.626 \times 10^{-34}}{4 \times 3.14 \times 0.01 \times 10^{-5}} \]

5. **Calculating the Denominator:**
First, calculate the denominator:
\[ 4 \times 3.14 \times 0.01 \times 10^{-5} = 1.256 \times 10^{-6} \]

6. **Final Calculation:**
Now, calculate Δv:
\[ \Delta v = \frac{6.626 \times 10^{-34}}{1.256 \times 10^{-6}} \approx 5.28 \times 10^{-28} \text{ m/s} \]

### Conclusion:
The uncertainty in velocity (Δv) is approximately \(5.28 \times 10^{-28}\) m/s.