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 heavier dust particle as compared to the lighter dust particle is

Options

A. 5 times
B. `(1)/(5)th`
C. same
D. 50 times

Explanation

To solve the problem of uncertainty in the velocity of the two dust particles, we can use Heisenberg's Uncertainty Principle, which states that:

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

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

Since momentum \(p\) is given by \(p = mv\), the uncertainty in momentum can be expressed as:

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

where:
- \(m\) is the mass of the particle,
- \(\Delta v\) is the uncertainty in velocity.

### Step 1: Identify the given values
- Mass of the lighter dust particle, \(m_1 = 0.1 \, \text{mg} = 0.1 \times 10^{-3} \, \text{g} = 0.1 \times 10^{-6} \, \text{kg}\)
- Mass of the heavier dust particle, \(m_2 = 0.5 \, \text{mg} = 0.5 \times 10^{-3} \, \text{g} = 0.5 \times 10^{-6} \, \text{kg}\)
- Uncertainty in position, \(\Delta x = 10^{-4} \, \text{m}\)

### Step 2: Calculate the uncertainty in velocity for both particles
Using the uncertainty principle:

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

Substituting for \(\Delta p\):

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

Rearranging gives:

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

### Step 3: Calculate \(\Delta v\) for the lighter particle
For the lighter particle (\(m_1\)):

\[ \Delta v_1 \geq \frac{h}{4\pi \Delta x m_1} \]

Substituting the values:

\[ \Delta v_1 \geq \frac{6.626 \times 10^{-34}}{4\pi (10^{-4}) (0.1 \times 10^{-6})} \]

Calculating this gives:

\[ \Delta v_1 \geq \frac{6.626 \times 10^{-34}}{4 \times 3.14 \times 10^{-4} \times 0.1 \times 10^{-6}} \]

### Step 4: Calculate \(\Delta v\) for the heavier particle
For the heavier particle (\(m_2\)):

\[ \Delta v_2 \geq \frac{h}{4\pi \Delta x m_2} \]

Substituting the values:

\[ \Delta v_2 \geq \frac{6.626 \times 10^{-34}}{4\pi (10^{-4}) (0.5 \times 10^{-6})} \]

Calculating this gives:

\[ \Delta v_2 \geq \frac{6.626 \times 10^{-34}}{4 \times 3.14 \times 10^{-4} \times 0.5 \times 10^{-6}} \]

### Step 5: Compare the uncertainties in velocity
To find the ratio of the uncertainties in velocity:

\[ \frac{\Delta v_2}{\Delta v_1} = \frac{m_1}{m_2} \]

Substituting the masses:

\[ \frac{\Delta v_2}{\Delta v_1} = \frac{0.1 \times 10^{-6}}{0.5 \times 10^{-6}} = \frac{0.1}{0.5} = \frac{1}{5} \]

Thus, the uncertainty in the velocity of the heavier dust particle is 5 times smaller than that of the lighter dust particle.

### Final Answer:
The uncertainty in the velocity of the heavier dust particle is 5 times smaller compared to the lighter dust particle.