Werner Heisenberg considered the limits of how precisely we can measure the properties of an electron or other microscopic particle. He determined that there is a fundamental limit to how closely we can measure both position and momentum. The more accurately we measure the momentum of a particle, the less accurately we can determine its position. The converse also true. This is summed up in what we now call the Heisenberg uncertainty principle.
The equation si `deltax.delta (mv)ge(h)/(4pi)`
The uncertainty in the position or in the momentum of a marcroscopic object like a baseball is too small to observe. However, the mass of microscopic object such as an electon is small enough for the uncertainty to be relatively large and significant.
If the uncertainties in position and momentum are equal, the uncertainty in the velocity is :

Options

A. `sqrt(h/pi)`
B. `sqrt(h/(2pi))`
C. `1/(2m)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 is expressed mathematically as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( \Delta x \) is the uncertainty in position, \( \Delta p \) is the uncertainty in momentum, and \( h \) is Planck's constant.

2. **Relate Momentum to Velocity**: The momentum \( p \) of an object is given by:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity. Thus, the uncertainty in momentum can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where \( \Delta v \) is the uncertainty in velocity.

3. **Set the Uncertainties Equal**: According to the problem, the uncertainties in position and momentum are equal:
\[ \Delta x = \Delta p \]
Therefore, we can substitute \( \Delta p \) with \( m \cdot \Delta v \):
\[ \Delta x = m \cdot \Delta v \]

4. **Substitute into the Uncertainty Principle**: Now substitute \( \Delta p \) in the Heisenberg equation:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

5. **Replace \( \Delta x \) with \( \Delta v \)**: Since \( \Delta x = \Delta v \), we can rewrite the equation:
\[ \Delta v \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]
This simplifies to:
\[ m \cdot \Delta v^2 \geq \frac{h}{4\pi} \]

6. **Isolate \( \Delta v^2 \)**: To find \( \Delta v^2 \), divide both sides by \( m \):
\[ \Delta v^2 \geq \frac{h}{4\pi m} \]

7. **Take the Square Root**: Finally, take the square root of both sides to find \( \Delta v \):
\[ \Delta v \geq \sqrt{\frac{h}{4\pi m}} \]

### Final Answer:
The uncertainty in velocity \( \Delta v \) is:
\[ \Delta v = \sqrt{\frac{h}{4\pi m}} \]