Johnny Jumper's favorite trick is to step out of his 16thstory window and fall 50.0 m into a pool. A news reporter takes a picture of 75.0-kg Johnny just before he makes a splash, using an exposure time of 5.00 ms. Find Johnny's de Broglie wavelength at this moment,
Johnny Jumper's favorite trick is to step out of his 16thstory window and fall 50.0 m into a pool. A news reporter takes a picture of 75.0-kg Johnny just before he makes a splash, using an exposure time of 5.00 ms. Find Johnny's de Broglie wavelength at this moment,
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![### Johnny Jumper's Free Fall and de Broglie Wavelength Calculation
**Scenario:**
Johnny Jumper's favorite trick is to step out of his 16th-story window and fall 50.0 m into a pool. A news reporter takes a picture of 75.0-kg Johnny just before he makes a splash, using an exposure time of 5.00 ms. Find Johnny’s de Broglie wavelength at this moment.
**Calculation Steps:**
1. **Determine Johnny's Velocity Before Impact:**
- Use the equation for free fall to determine the velocity (v) just before impact.
\[
v = \sqrt{2 \cdot g \cdot h}
\]
- \(g = 9.81 \, \text{m/s}^2\) (acceleration due to gravity)
- \(h = 50.0 \, \text{m}\)
\[
v = \sqrt{2 \cdot 9.81 \, \text{m/s}^2 \cdot 50.0 \, \text{m}} = \sqrt{981} \approx 31.3 \, \text{m/s}
\]
2. **Calculate de Broglie Wavelength:**
- The de Broglie wavelength (\(\lambda\)) can be calculated using the formula:
\[
\lambda = \frac{h}{p}
\]
- where \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \, \text{Js}\)) and \(p\) is momentum.
- Momentum \(p\) is given by:
\[
p = m \cdot v
\]
- \(m = 75.0 \, \text{kg}\)
\[
p = 75.0 \, \text{kg} \times 31.3 \, \text{m/s} \approx 2347.5 \, \text{kg} \cdot \text{m/s}
\]
- Therefore, the de Broglie wavelength is:
\[
\lambda = \frac{6.626 \times 10^{-34} \, \text{Js}}{2347.5 \, \text{kg} \cd](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fe8677b-d2b4-4cf1-b1ed-08820154fcb5%2F78017080-8079-4385-9184-298bc99172d8%2Fyn9upjp_processed.png&w=3840&q=75)
Transcribed Image Text:### Johnny Jumper's Free Fall and de Broglie Wavelength Calculation
**Scenario:**
Johnny Jumper's favorite trick is to step out of his 16th-story window and fall 50.0 m into a pool. A news reporter takes a picture of 75.0-kg Johnny just before he makes a splash, using an exposure time of 5.00 ms. Find Johnny’s de Broglie wavelength at this moment.
**Calculation Steps:**
1. **Determine Johnny's Velocity Before Impact:**
- Use the equation for free fall to determine the velocity (v) just before impact.
\[
v = \sqrt{2 \cdot g \cdot h}
\]
- \(g = 9.81 \, \text{m/s}^2\) (acceleration due to gravity)
- \(h = 50.0 \, \text{m}\)
\[
v = \sqrt{2 \cdot 9.81 \, \text{m/s}^2 \cdot 50.0 \, \text{m}} = \sqrt{981} \approx 31.3 \, \text{m/s}
\]
2. **Calculate de Broglie Wavelength:**
- The de Broglie wavelength (\(\lambda\)) can be calculated using the formula:
\[
\lambda = \frac{h}{p}
\]
- where \(h\) is Planck’s constant (\(6.626 \times 10^{-34} \, \text{Js}\)) and \(p\) is momentum.
- Momentum \(p\) is given by:
\[
p = m \cdot v
\]
- \(m = 75.0 \, \text{kg}\)
\[
p = 75.0 \, \text{kg} \times 31.3 \, \text{m/s} \approx 2347.5 \, \text{kg} \cdot \text{m/s}
\]
- Therefore, the de Broglie wavelength is:
\[
\lambda = \frac{6.626 \times 10^{-34} \, \text{Js}}{2347.5 \, \text{kg} \cd
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