The uncertainty in the velocity of particle of mass `6.626xx10^(-28)`kg is `10^(-6)` m/sec. What is the uncertaintly in its position in m?

Options

A. `(1)/(2pi)`
B. `(2.5)/(pi)`
C. `(4)/(pi)`
D. `(1)/(4pi)`

Explanation

To solve the problem of finding the uncertainty in the position of a particle given its mass and the uncertainty in its velocity, we will use the Heisenberg Uncertainty Principle. Here’s a step-by-step solution:

### Step 1: Understand the Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that:
\[ \Delta p \cdot \Delta x \geq \frac{h}{4\pi} \]
where:
- \(\Delta p\) is the uncertainty in momentum,
- \(\Delta x\) is the uncertainty in position,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J s}\)).

### Step 2: Relate momentum to mass and velocity
Momentum (\(p\)) is given by the formula:
\[ p = mv \]
Thus, the uncertainty in momentum (\(\Delta p\)) 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 3: Substitute into the uncertainty principle
Substituting \(\Delta p\) into the Heisenberg Uncertainty Principle gives:
\[ m \cdot \Delta v \cdot \Delta x \geq \frac{h}{4\pi} \]

### Step 4: Solve for \(\Delta x\)
Rearranging the equation to solve for the uncertainty in position (\(\Delta x\)):
\[ \Delta x \geq \frac{h}{4\pi m \Delta v} \]

### Step 5: Plug in the values
Given:
- Mass \(m = 6.626 \times 10^{-28} \, \text{kg}\),
- Uncertainty in velocity \(\Delta v = 10^{-6} \, \text{m/s}\),
- Planck's constant \(h = 6.626 \times 10^{-34} \, \text{J s}\).

Now substituting these values into the equation:
\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4\pi (6.626 \times 10^{-28})(10^{-6})} \]

### Step 6: Simplify the expression
Calculating the denominator:
\[ 4\pi (6.626 \times 10^{-28})(10^{-6}) = 4\pi \cdot 6.626 \times 10^{-34} \]
Thus:
\[ \Delta x \geq \frac{6.626 \times 10^{-34}}{4\pi \cdot 6.626 \times 10^{-34}} = \frac{1}{4\pi} \]

### Step 7: Final result
The uncertainty in position is:
\[ \Delta x \geq \frac{1}{4\pi} \, \text{m} \]

### Conclusion
The uncertainty in the position of the particle is \(\frac{1}{4\pi} \, \text{m}\).

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