It is not possible to determine precisely both the position and momentum (or velocity) of a small moving particle such as electron, proton etc. This is known as Heisenberg uncertainty principle. The mathematical form of this principle is :
`Delta x.Delta p ge (h)/(4pi)` (constant)
However this principle is irrelevant in case of bigger particles such as a cup, ball, car etc., that we come across in our daily life.
If uncertainty in position and momentum are equal, the v uncertainty in velocity would be

Options

A. `(1)/(2m)sqrt((h)/(pi))`
B. `sqrt((h)/(2pi))`
C. `sqrt((h)/(pi))`
D. None of these

Explanation

To solve the problem regarding the Heisenberg uncertainty principle and the uncertainty in velocity when the uncertainties in position and momentum are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Heisenberg Uncertainty Principle**:
The principle states that the product of the uncertainties in position (Δx) and momentum (Δp) is greater than or equal to a constant:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

2. **Set the Uncertainties Equal**:
If the uncertainties in position and momentum are equal, we can denote them as:
\[ \Delta x = \Delta p \]
Therefore, we can substitute Δp with Δx in the uncertainty principle:
\[ \Delta x \cdot \Delta x \geq \frac{h}{4\pi} \]
This simplifies to:
\[ (\Delta x)^2 \geq \frac{h}{4\pi} \]

3. **Express Δp in Terms of Δx**:
Since Δp = Δx, we can write:
\[ \Delta p = \Delta x \]

4. **Relate Momentum to Velocity**:
Momentum (p) is defined as the product of mass (m) and velocity (v):
\[ p = mv \]
Therefore, the uncertainty in momentum can be expressed as:
\[ \Delta p = m \Delta v \]
where Δv is the uncertainty in velocity.

5. **Substitute Δp in the Uncertainty Principle**:
Now substituting Δp in the uncertainty principle:
\[ \Delta x \cdot (m \Delta v) \geq \frac{h}{4\pi} \]
Simplifying gives:
\[ m \Delta x \Delta v \geq \frac{h}{4\pi} \]

6. **Solve for Δv**:
Rearranging the equation to solve for Δv:
\[ \Delta v \geq \frac{h}{4\pi m \Delta x} \]

7. **Substituting Δx**:
From step 2, we know that:
\[ \Delta x \geq \sqrt{\frac{h}{4\pi}} \]
Substituting this into the equation for Δv:
\[ \Delta v \geq \frac{h}{4\pi m \sqrt{\frac{h}{4\pi}}} \]
Simplifying this gives:
\[ \Delta v \geq \frac{h}{4\pi m} \cdot \frac{2\sqrt{\pi}}{\sqrt{h}} = \frac{1}{2m} \sqrt{\frac{h}{\pi}} \]

### Final Result:
Thus, the uncertainty in velocity (Δv) is:
\[ \Delta v \geq \frac{1}{2m} \sqrt{\frac{h}{\pi}} \]