0.5 gm particle has uncertainity of `2xx10^(-5)` m find the uncertainty in its velocity`(m//s)`

Options

A. `3.0xx10^(33)`
B. `5xx10^(-27)`
C. `4xx10^(-19)`
D. `4xx10^(-10)`

Explanation

To solve the problem of finding the uncertainty in the velocity of a 0.5 gm particle with a given uncertainty in position, we will use Heisenberg's Uncertainty Principle. The principle states that the product of the uncertainties in position (Δx) and momentum (Δp) is on the order of Planck's constant (h).

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the particle (m) = 0.5 gm = 0.5 × 10^(-3) kg (since 1 gm = 10^(-3) kg)
- Uncertainty in position (Δx) = 2 × 10^(-5) m

2. **Recall Heisenberg's Uncertainty Principle:**
The principle can be expressed as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where Δp is the uncertainty in momentum, and h is Planck's constant (h ≈ 6.626 × 10^(-34) Js).

3. **Relate momentum to velocity:**
Momentum (p) is given by the product of mass and velocity:
\[ p = m \cdot v \]
Therefore, the uncertainty in momentum (Δp) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where Δv is the uncertainty in velocity.

4. **Substituting into the uncertainty principle:**
We can rewrite the uncertainty principle as:
\[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]

5. **Rearranging to find Δv:**
\[ \Delta v \geq \frac{h}{4\pi m \Delta x} \]

6. **Substituting the known values:**
- h = 6.626 × 10^(-34) Js
- m = 0.5 × 10^(-3) kg
- Δx = 2 × 10^(-5) m

Plugging in these values:
\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{4 \pi (0.5 \times 10^{-3}) (2 \times 10^{-5})} \]

7. **Calculating the denominator:**
- First calculate \(4 \pi (0.5 \times 10^{-3}) (2 \times 10^{-5})\):
\[ 4 \pi \approx 12.566 \]
\[ 4 \pi (0.5 \times 10^{-3}) (2 \times 10^{-5}) \approx 12.566 \times 0.5 \times 10^{-3} \times 2 \times 10^{-5} \approx 1.2566 \times 10^{-7} \]

8. **Final calculation for Δv:**
\[ \Delta v \geq \frac{6.626 \times 10^{-34}}{1.2566 \times 10^{-7}} \approx 5.28 \times 10^{-27} \text{ m/s} \]

9. **Conclusion:**
The uncertainty in the velocity of the particle is approximately \(5.28 \times 10^{-27} \text{ m/s}\).