Question
Compare the wavelength of of an electron traveling st 3.00e6 m/s with that of a baseball of mass 0.145kg having a speed of 26.8m/s( for an electron m=9.019e-31kg
Answers
GPT-4o mini
To compare the wavelengths of an electron and a baseball, we can use the de Broglie wavelength formula, which is given by:
\[
\lambda = \frac{h}{p}
\]
where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\)),
- \(p\) is the momentum of the particle, defined as \(p = mv\), where \(m\) is the mass and \(v\) is the velocity of the particle.
### Step 1: Calculate the wavelength of the electron
1. **Mass of the electron, \(m_e\)**:
\[
m_e = 9.019 \times 10^{-31} \text{ kg}
\]
2. **Velocity of the electron, \(v_e\)**:
\[
v_e = 3.00 \times 10^6 \text{ m/s}
\]
3. **Calculate the momentum of the electron**:
\[
p_e = m_e v_e = (9.019 \times 10^{-31} \text{ kg})(3.00 \times 10^6 \text{ m/s}) = 2.7057 \times 10^{-24} \text{ kg m/s}
\]
4. **Calculate the wavelength of the electron**:
\[
\lambda_e = \frac{h}{p_e} = \frac{6.626 \times 10^{-34} \text{ J s}}{2.7057 \times 10^{-24} \text{ kg m/s}} \approx 2.448 \times 10^{-10} \text{ m}
\]
### Step 2: Calculate the wavelength of the baseball
1. **Mass of the baseball, \(m_b\)**:
\[
m_b = 0.145 \text{ kg}
\]
2. **Velocity of the baseball, \(v_b\)**:
\[
v_b = 26.8 \text{ m/s}
\]
3. **Calculate the momentum of the baseball**:
\[
p_b = m_b v_b = (0.145 \text{ kg})(26.8 \text{ m/s}) = 3.886 \text{ kg m/s}
\]
4. **Calculate the wavelength of the baseball**:
\[
\lambda_b = \frac{h}{p_b} = \frac{6.626 \times 10^{-34} \text{ J s}}{3.886 \text{ kg m/s}} \approx 1.703 \times 10^{-34} \text{ m}
\]
### Step 3: Summary of Results
- **Wavelength of the electron**: \(\lambda_e \approx 2.448 \times 10^{-10} \text{ m}\)
- **Wavelength of the baseball**: \(\lambda_b \approx 1.703 \times 10^{-34} \text{ m}\)
### Conclusion
The wavelength of the electron (approximately \(2.448 \times 10^{-10} \text{ m}\)) is significantly larger than that of the baseball (approximately \(1.703 \times 10^{-34} \text{ m}\)). This illustrates the wave-particle duality effect being more pronounced at the quantum scale (for the electron) than at the macroscopic scale (for the baseball).
\[
\lambda = \frac{h}{p}
\]
where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ J s}\)),
- \(p\) is the momentum of the particle, defined as \(p = mv\), where \(m\) is the mass and \(v\) is the velocity of the particle.
### Step 1: Calculate the wavelength of the electron
1. **Mass of the electron, \(m_e\)**:
\[
m_e = 9.019 \times 10^{-31} \text{ kg}
\]
2. **Velocity of the electron, \(v_e\)**:
\[
v_e = 3.00 \times 10^6 \text{ m/s}
\]
3. **Calculate the momentum of the electron**:
\[
p_e = m_e v_e = (9.019 \times 10^{-31} \text{ kg})(3.00 \times 10^6 \text{ m/s}) = 2.7057 \times 10^{-24} \text{ kg m/s}
\]
4. **Calculate the wavelength of the electron**:
\[
\lambda_e = \frac{h}{p_e} = \frac{6.626 \times 10^{-34} \text{ J s}}{2.7057 \times 10^{-24} \text{ kg m/s}} \approx 2.448 \times 10^{-10} \text{ m}
\]
### Step 2: Calculate the wavelength of the baseball
1. **Mass of the baseball, \(m_b\)**:
\[
m_b = 0.145 \text{ kg}
\]
2. **Velocity of the baseball, \(v_b\)**:
\[
v_b = 26.8 \text{ m/s}
\]
3. **Calculate the momentum of the baseball**:
\[
p_b = m_b v_b = (0.145 \text{ kg})(26.8 \text{ m/s}) = 3.886 \text{ kg m/s}
\]
4. **Calculate the wavelength of the baseball**:
\[
\lambda_b = \frac{h}{p_b} = \frac{6.626 \times 10^{-34} \text{ J s}}{3.886 \text{ kg m/s}} \approx 1.703 \times 10^{-34} \text{ m}
\]
### Step 3: Summary of Results
- **Wavelength of the electron**: \(\lambda_e \approx 2.448 \times 10^{-10} \text{ m}\)
- **Wavelength of the baseball**: \(\lambda_b \approx 1.703 \times 10^{-34} \text{ m}\)
### Conclusion
The wavelength of the electron (approximately \(2.448 \times 10^{-10} \text{ m}\)) is significantly larger than that of the baseball (approximately \(1.703 \times 10^{-34} \text{ m}\)). This illustrates the wave-particle duality effect being more pronounced at the quantum scale (for the electron) than at the macroscopic scale (for the baseball).