From the previous DeBroglie example, compare the velocity uncertainty measurement in both Dr. Ochoa and the electron if the uncertainty in position, \( \Delta x \), is \( 1.00 \times 10^{-12} \mathrm{~m} \). Note: The mass of Dr. Ochoa was 84.55 kg and he was running at \( 2.68 \mathrm{~m} / \mathrm{s} \), the electron has a mass of \( 9.109 \times 10^{-31} \mathrm{~kg} \) and was moving at \( 1.00 \times 10^{8} \mathrm{~m} / \mathrm{s} \).

To compare the velocity uncertainty measurements for Dr. Ochoa and the electron, we can use the Heisenberg uncertainty principle, which states: \[ \Delta x \Delta p \geq \frac{\hbar}{2} \] where: - \(\Delta x\) is the uncertainty in position, - \(\Delta p\) is the uncertainty in momentum, - \(\hbar\) is the reduced Planck's constant, approximately \(1.055 \times 10^{-34} \, \text{Js}\). The momentum \(p\) of an object is given by: \[ p = mv \] where \(m\) is the mass and \(v\) is the velocity. The uncertainty in momentum can be expressed as: \[ \Delta p = m \Delta v \] where \(\Delta v\) is the uncertainty in velocity. Therefore, we can rewrite the uncertainty principle in terms of velocity: \[ \Delta x \cdot m \Delta v \geq \frac{\hbar}{2} \] From this, we can solve for \(\Delta v\): \[ \Delta v \geq \frac{\hbar}{2 m \Delta x} \] Now, we can calculate \(\Delta v\) for both Dr. Ochoa and the electron. ### For Dr. Ochoa: 1. **Mass**: \(m_O = 84.55 \, \text{kg}\) 2. **Uncertainty in position**: \(\Delta x = 1.00 \times 10^{-12} \, \text{m}\) 3. **Reduced Planck's constant**: \(\hbar \approx 1.055 \times 10^{-34} \, \text{Js}\) Now, substituting these values into the equation for \(\Delta v\): \[ \Delta v_O \geq \frac{1.055 \times 10^{-34}}{2 \times 84.55 \times 1.00 \times 10^{-12}} \] Calculating the denominator: \[ 2 \times 84.55 \times 1.00 \times 10^{-12} = 1.691 \times 10^{-10} \] Now, calculating \(\Delta v_O\): \[ \Delta v_O \geq \frac{1.055 \times 10^{-34}}{1.691 \times 10^{-10}} \approx 6.24 \times 10^{-25} \, \text{m/s} \] ### For the Electron: 1. **Mass**: \(m_e = 9.109 \times 10^{-31} \, \text{kg}\) 2. **Uncertainty in position**: \(\Delta x = 1.00 \times 10^{-12} \, \text{m}\) Now, substituting these values into the equation for \(\Delta v\): \[ \Delta v_e \geq \frac{1.055 \times 10^{-34}}{2 \times 9.109 \times 10^{-31} \times 1.00 \times 10^{-12}} \] Calculating the denominator: \[ 2 \times 9.109 \times 10^{-31} \times 1.00 \times 10^{-12} = 1.822 \times 10^{-42} \] Now, calculating \(\Delta v_e\): \[ \Delta v_e \geq \frac{1.055 \times 10^{-34}}{1.822 \times 10^{-42}} \approx 5.79 \times 10^{7} \, \text{m/s} \] ### Summary of Results: - **Uncertainty in velocity for Dr. Ochoa**: \(\Delta v_O \geq 6.24 \times 10^{-25} \, \text{m/s}\) - **Uncertainty in velocity for the electron**: \(\Delta v_e \geq 5.79 \times 10^{7} \, \text{m/s}\) ### Conclusion: The uncertainty in velocity for the electron is vastly greater than that for Dr. Ochoa, which is consistent with the principles of quantum mechanics, where smaller masses (like that of an electron) exhibit much larger uncertainties in their velocity when the position is measured with a certain precision.