The uncertainty in the position of a flying dust particle of mass 0.1 mg is found to be `10^(-4)`m. The uncertainty in the position of another flying dust particle of mass 0.5 mg is also found to be same
The uncertainty in the velocity of the lighter dust particle is nearly

Options

A. `5 xx 10^(-24) ms^(-1)`
B. `2.5 xx 10^(-24) ms^(-1)`
C. `1.25 xx 10^(-24) ms^(-1)`
D. `5 xx 10^(-25) ms^(-1)`

Explanation

To solve the problem, we will use the Heisenberg Uncertainty Principle, which states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is given by:

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

Where:
- \( \Delta x \) is the uncertainty in position,
- \( \Delta p \) is the uncertainty in momentum,
- \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \, \text{Js} \).

Since momentum \( p \) is given by the product of mass \( m \) and velocity \( v \) (i.e., \( p = mv \)), we can express the uncertainty in momentum as:

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

Where:
- \( \Delta v \) is the uncertainty in velocity.

Now, we can substitute this into the uncertainty principle equation:

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

### Step 1: Identify the given values
- Mass of the lighter dust particle, \( m = 0.1 \, \text{mg} = 0.1 \times 10^{-3} \, \text{g} = 0.1 \times 10^{-6} \, \text{kg} = 1 \times 10^{-4} \, \text{kg} \)
- Uncertainty in position, \( \Delta x = 10^{-4} \, \text{m} \)
- Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{Js} \)

### Step 2: Substitute the values into the equation
Substituting the known values into the uncertainty principle equation:

\[ 10^{-4} \cdot (1 \times 10^{-4} \cdot \Delta v) \geq \frac{6.626 \times 10^{-34}}{4\pi} \]

### Step 3: Calculate the right side of the inequality
Calculating \( \frac{h}{4\pi} \):

\[ \frac{6.626 \times 10^{-34}}{4 \cdot 3.14} \approx \frac{6.626 \times 10^{-34}}{12.56} \approx 5.28 \times 10^{-35} \, \text{Js} \]

### Step 4: Solve for \( \Delta v \)
Now we can solve for \( \Delta v \):

\[ 10^{-4} \cdot (1 \times 10^{-4} \cdot \Delta v) \geq 5.28 \times 10^{-35} \]

This simplifies to:

\[ 10^{-8} \cdot \Delta v \geq 5.28 \times 10^{-35} \]

Now, dividing both sides by \( 10^{-8} \):

\[ \Delta v \geq \frac{5.28 \times 10^{-35}}{10^{-8}} = 5.28 \times 10^{-27} \, \text{m/s} \]

### Conclusion
The uncertainty in the velocity of the lighter dust particle is approximately \( 5.28 \times 10^{-27} \, \text{m/s} \).