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 (\( \Delta v \)) of a moving bullet given the uncertainty in position (\( \Delta x \)), we can use Heisenberg's uncertainty principle. The principle states that the product of the uncertainties in position and momentum is always greater than or equal to \( \frac{h}{4\pi} \), where \( h \) is Planck's constant.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Uncertainty in position (\( \Delta x \)) = \( 10^{-5} \, m \)
- Mass of the bullet (\( m \)) = \( 10 \, gm = 10 \times 10^{-3} \, kg = 0.01 \, kg \)
- Planck's constant (\( h \)) = \( 6.626 \times 10^{-34} \, Js \)

2. **Use Heisenberg's Uncertainty Principle:**
The principle can be expressed as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( \Delta p \) is the uncertainty in momentum.

3. **Relate Momentum to Velocity:**
The momentum (\( p \)) is given by:
\[ p = m \cdot v \]
Therefore, the uncertainty in momentum (\( \Delta p \)) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
Substituting this into the uncertainty principle gives:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

4. **Rearranging the Equation:**
We can rearrange the equation to solve for \( \Delta v \):
\[ \Delta v \geq \frac{h}{4\pi \Delta x \cdot m} \]

5. **Substituting the Known Values:**
Plugging in the values:
\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{4 \cdot \pi \cdot (10^{-5}) \cdot (0.01)} \]

6. **Calculating the Denominator:**
First, calculate \( 4 \cdot \pi \cdot (10^{-5}) \cdot (0.01) \):
\[ 4 \cdot \pi \approx 12.566 \]
\[ 12.566 \cdot 10^{-7} = 1.2566 \times 10^{-6} \]

7. **Final Calculation:**
Now calculate \( \Delta v \):
\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{1.2566 \times 10^{-6}} \approx 5.28 \times 10^{-28} \, m/s \]

### Conclusion:
The uncertainty in velocity (\( \Delta v \)) of the moving bullet is approximately:
\[ \Delta v \approx 5.28 \times 10^{-28} \, m/s \]