According to the Heisenberg uncertainty principle, there is a fundamental limit to how precisely we can simultaneously know the position and momentum (or speed) of a particle such as an electron. The principle states that the product of the uncertainties in these two quantities must be greater than or equal to a certain value.
To calculate the uncertainty in the position of an electron given the uncertainty in its speed, we can use the following formula:
\[\Delta x \cdot \Delta v \geq \frac{\hbar}{2}\]
Where:
- \(\Delta x\) represents the uncertainty in position,
- \(\Delta v\) represents the uncertainty in velocity (speed),
- \(\hbar\) is the reduced Planck's constant (approximately \(1.054 \times 10^{-34} \, \text{J} \cdot \text{s}\)).
Given that the uncertainty in speed (\(\Delta v\)) is \(2.30 \times 10^{3} \, \text{m/s}\), we can substitute this value into the equation to find the uncertainty in position (\(\Delta x\)).
\[\Delta x \cdot (2.30 \times 10^{3} \, \text{m/s}) \geq \frac{1.054 \times 10^{-34} \, \text{J} \cdot \text{s}}{2}\]
Now we can solve for \(\Delta x\):
\[\Delta x \geq \frac{1.054 \times 10^{-34} \, \text{J} \cdot \text{s}}{2 \cdot (2.30 \times 10^{3} \, \text{m/s})}\]
\[\Delta x \geq 2.293 \times 10^{-38} \, \text{m}\]
Therefore, the uncertainty in the position of the electron is greater than or equal to \(2.293 \times 10^{-38} \, \text{m}\). This means that we cannot know the exact position of the electron more precisely than this value due to the inherent limitations imposed by the Heisenberg uncertainty principle.