Uncertainty in position of a particle of `25g` in space is `10^(8)m` Hence uncertainty in velocity `(ms^(-1))` is (Planck's constant `h=6.6xx10^(-34)Js`)

Options

A. `2.1xx10^(-28)`
B. `2.1xx10^(-34)`
C. `0.5xx10^(-34)`
D. `5.0xx20^(-24)`

Answer

To find the uncertainty in velocity (Δv) of a particle with a given uncertainty in position (Δx), we can use the Heisenberg Uncertainty Principle. The principle states that:

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

Where:
- Δx = uncertainty in position
- Δp = uncertainty in momentum
- h = Planck's constant

Since momentum (p) is defined as the product of mass (m) and velocity (v), we can express the uncertainty in momentum as:

\[ \Delta p = m \cdot \Delta v \]

Substituting this into the uncertainty principle gives us:

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

From this, we can solve for the uncertainty in velocity (Δv):

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

Now, we can substitute the known values into this equation.

1. **Convert mass from grams to kilograms:**
\[ m = 25 \, \text{g} = 25 \times 10^{-3} \, \text{kg} = 0.025 \, \text{kg} \]

2. **Substitute the values into the equation:**
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{Js} \)
- Uncertainty in position \( \Delta x = 10^8 \, \text{m} \)

Now substituting these values into the equation for Δv:

\[ \Delta v = \frac{6.626 \times 10^{-34}}{4 \pi (0.025)(10^8)} \]

3. **Calculate the denominator:**
- First calculate \( 4 \pi \):
\[ 4 \pi \approx 12.566 \]
- Now calculate \( 4 \pi m \Delta x \):
\[ 4 \pi (0.025)(10^8) = 12.566 \times 0.025 \times 10^8 \approx 3.1415 \times 10^6 \]

4. **Calculate Δv:**
\[ \Delta v = \frac{6.626 \times 10^{-34}}{3.1415 \times 10^6} \approx 2.11 \times 10^{-28} \, \text{m/s} \]

Thus, the uncertainty in velocity (Δv) is approximately:

\[ \Delta v \approx 2.1 \times 10^{-28} \, \text{m/s} \]

### Final Answer:
The uncertainty in velocity is \( 2.1 \times 10^{-28} \, \text{m/s} \).

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