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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
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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