If uncertainty in position are velocity are equal the uncertainty in momentum will be

Options

A. `(1)/(2)sqrt((mh)/(pi))`
B. `(1)/(2)sqrt((h)/(pi m))`
C. `(h)/(4pi m)`
D. `(mh)/(4pi)`

Explanation

To solve the problem, we need to understand the relationship between uncertainty in position (Δx), uncertainty in velocity (Δv), and uncertainty in momentum (Δp).

### Step-by-Step Solution:

1. **Understanding Uncertainty Principle**: According to Heisenberg's Uncertainty Principle, the product of the uncertainties in position and momentum is given by:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \(h\) is Planck's constant.

2. **Relating Momentum and Velocity**: Momentum (p) is defined as:
\[ p = mv \]
where \(m\) is mass and \(v\) is velocity. The uncertainty in momentum (Δp) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]

3. **Substituting Δp in the Uncertainty Principle**: By substituting Δp in the uncertainty principle, we get:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

4. **Rearranging the Equation**: We can rearrange this equation to express the relationship between Δx and Δv:
\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]

5. **Conclusion**: If the uncertainty in position (Δx) and the uncertainty in velocity (Δv) are equal, we can denote them as Δx = Δv. Therefore, we can write:
\[ \Delta x \cdot \Delta x \geq \frac{h}{4\pi m} \]
This implies:
\[ (\Delta x)^2 \geq \frac{h}{4\pi m} \]
Thus, the uncertainty in momentum (Δp) will be equal to \(m \cdot \Delta v\) and can be expressed in terms of Δx.

### Final Answer:
The uncertainty in momentum (Δp) will be equal to \(m \cdot \Delta v\), and if Δx = Δv, then:
\[ \Delta p = m \cdot \Delta x \]