Uncertainty in position of a particle of 25 g in space is `10^(-5)` m. Hence, uncertainty in velocity `(m s^(-1))` is (Planck's constant `h=6.6 xx 10^(-34)Js)`

Options

A. `2.1xx 10^(-28)`
B. `2.1xx 10^(-34)`
C. `0.5xx 10^(-34)`
D. `5.0xx 10^(-24)`

Explanation

To solve the problem of finding the uncertainty in velocity given the uncertainty in position, we can use Heisenberg's Uncertainty Principle, which states:

\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]

Where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta v\) is the uncertainty in velocity,
- \(h\) is Planck's constant,
- \(m\) is the mass of the particle.

### Step-by-Step Solution:

1. **Convert the mass from grams to kilograms**:
The mass of the particle is given as 25 g. We need to convert this to kilograms.
\[ m = 25 \, \text{g} = 25 \times 10^{-3} \, \text{kg} = 0.025 \, \text{kg} \]

**Hint**: Remember to convert grams to kilograms by dividing by 1000.

2. **Identify the given values**:
- Uncertainty in position, \(\Delta x = 10^{-5} \, \text{m}\)
- Planck's constant, \(h = 6.6 \times 10^{-34} \, \text{Js}\)

3. **Use the formula to find uncertainty in velocity**:
Rearranging the Heisenberg's equation to find \(\Delta v\):
\[ \Delta v = \frac{h}{4\pi m \Delta x} \]

4. **Substitute the values into the equation**:
\[ \Delta v = \frac{6.6 \times 10^{-34}}{4 \times 3.14 \times (0.025) \times (10^{-5})} \]

5. **Calculate the denominator**:
First, calculate \(4 \times 3.14 \times 0.025 \times 10^{-5}\):
\[ 4 \times 3.14 = 12.56 \]
\[ 12.56 \times 0.025 = 0.314 \]
\[ 0.314 \times 10^{-5} = 3.14 \times 10^{-6} \]

6. **Now substitute back to find \(\Delta v\)**:
\[ \Delta v = \frac{6.6 \times 10^{-34}}{3.14 \times 10^{-6}} \]

7. **Perform the division**:
\[ \Delta v = 2.1 \times 10^{-28} \, \text{m/s} \]

8. **Final answer**:
The uncertainty in velocity is:
\[ \Delta v \approx 2.1 \times 10^{-28} \, \text{m/s} \]

### Conclusion:
The correct option for the uncertainty in velocity is \(2.1 \times 10^{-28} \, \text{m/s}\).

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