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PRACTICE IT Use the worked example above to help you solve this problem. (a) Compare the de Broglie wavelength for an electron (m¸ = 9.11 x 10 31 kg) moving at a speed of 1.21 x 107 m/s with that of a baseball of mass 0.145 kg pitched at 43.7 m/s. 2 = 5.987E-11 m 1.04E-34 (b) Compare these wavelengths with that of an electron traveling at 0.999c. m

PRACTICE IT Use the worked example above to help you solve this problem. (a) Compare the de Broglie wavelength for an electron (m¸ = 9.11 x 10 31 kg) moving at a speed of 1.21 x 107 m/s with that of a baseball of mass 0.145 kg pitched at 43.7 m/s. 2 = 5.987E-11 m 1.04E-34 (b) Compare these wavelengths with that of an electron traveling at 0.999c. m

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EXAMPLE 27.4 The Electron Versus the Baseball GOAL Apply the de Broglie hypothesis to a quantum and a classical object. PROBLEM (a) Compare the de Broglie wavelength for an electron (m, = 9.11 x 10-31 kg) moving at a speed equal to 1.0 x 107 m/s with that of a baseball of mass 0.145 kg pitched at 45.0 m/s. (b) Compare these wavelengths with that of an electron traveling at 0.999c. STRATEGY This problem is a matter of substitution into the equation for the de Broglie wavelength. In part (b) the relativistic momentum must be used. SOLUTION (A) Compare the de Broglie wavelengths of the electron and the baseball. Substitute data for the electron into h 6.63 x 10-34 J.S the De Broglie wavelength equation: 7.28 x 10-11 m (9.11 x 10-31 kg)(1.00 x 107 m/s) Repeat the calculation with the h 6.63 x 10-34 J•s baseball data: 1.02 x 10-34 m = m;v (0.145 kg)(45.0 m/s) (B) Find the wavelength for an electron traveling at 0.999c. Replace the momentum in de Broglie's equation with the relativistic hv1- v?/c2 h m_y/V1- v²/c? momentum: Substitute: (6.63 x 1034 J.•s)V1 - (0.999c)²/c² (9.11 x 10 31 kg)(0.999 · 3.00 × 108 m/s) = 1.09 x 10-13 m
Transcribed Image Text:EXAMPLE 27.4 The Electron Versus the Baseball GOAL Apply the de Broglie hypothesis to a quantum and a classical object. PROBLEM (a) Compare the de Broglie wavelength for an electron (m, = 9.11 x 10-31 kg) moving at a speed equal to 1.0 x 107 m/s with that of a baseball of mass 0.145 kg pitched at 45.0 m/s. (b) Compare these wavelengths with that of an electron traveling at 0.999c. STRATEGY This problem is a matter of substitution into the equation for the de Broglie wavelength. In part (b) the relativistic momentum must be used. SOLUTION (A) Compare the de Broglie wavelengths of the electron and the baseball. Substitute data for the electron into h 6.63 x 10-34 J.S the De Broglie wavelength equation: 7.28 x 10-11 m (9.11 x 10-31 kg)(1.00 x 107 m/s) Repeat the calculation with the h 6.63 x 10-34 J•s baseball data: 1.02 x 10-34 m = m;v (0.145 kg)(45.0 m/s) (B) Find the wavelength for an electron traveling at 0.999c. Replace the momentum in de Broglie's equation with the relativistic hv1- v?/c2 h m_y/V1- v²/c? momentum: Substitute: (6.63 x 1034 J.•s)V1 - (0.999c)²/c² (9.11 x 10 31 kg)(0.999 · 3.00 × 108 m/s) = 1.09 x 10-13 m
LEARN MORE REMARKS The electron wavelength corresponds to that of x-rays in the electromagnetic spectrum. The baseball, by contrast, has a wavelength much smaller than any aperture through which the baseball could possibly pass, so we couldn't observe any of its diffraction effects. It is generally true that the wave properties of large-scale objects can't be observed. Notice that even at extreme relativistic speeds, the electron wavelength is still far larger than the baseball's. QUESTION How does doubling the speed of a particle affect its wavelength? (Select all that apply.) V It reduces the wavelength to less than half at relativistic speeds where v/c gets close to 1. O At ordinary speeds where v/c << 1, it doubles the wavelength of the particle. O It doubles the wavelength of the particle at all speeds. It less than doubles the wavelength at relativistic speeds where v/c gets close to 1. O It halves the wavelength at all speeds. V At ordinary speeds where v/c << 1, it halves the wavelength of the particle. PRACTICE IT Use the worked example above to help you solve this problem. (a) Compare the de Broglie wavelength for an electron (m, = 9.11 x 10 31 kg) moving at a speed of 1.21 x 107 m/s with that of a baseball of mass 0.145 kg pitched at 43.7 m/s. = 5.987E-11 1.04E-34 (b) Compare these wavelengths with that of an electron traveling at 0.999c. m ..................... .....
Transcribed Image Text:LEARN MORE REMARKS The electron wavelength corresponds to that of x-rays in the electromagnetic spectrum. The baseball, by contrast, has a wavelength much smaller than any aperture through which the baseball could possibly pass, so we couldn't observe any of its diffraction effects. It is generally true that the wave properties of large-scale objects can't be observed. Notice that even at extreme relativistic speeds, the electron wavelength is still far larger than the baseball's. QUESTION How does doubling the speed of a particle affect its wavelength? (Select all that apply.) V It reduces the wavelength to less than half at relativistic speeds where v/c gets close to 1. O At ordinary speeds where v/c << 1, it doubles the wavelength of the particle. O It doubles the wavelength of the particle at all speeds. It less than doubles the wavelength at relativistic speeds where v/c gets close to 1. O It halves the wavelength at all speeds. V At ordinary speeds where v/c << 1, it halves the wavelength of the particle. PRACTICE IT Use the worked example above to help you solve this problem. (a) Compare the de Broglie wavelength for an electron (m, = 9.11 x 10 31 kg) moving at a speed of 1.21 x 107 m/s with that of a baseball of mass 0.145 kg pitched at 43.7 m/s. = 5.987E-11 1.04E-34 (b) Compare these wavelengths with that of an electron traveling at 0.999c. m ..................... .....
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