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Answer :
Final Answer:
The velocity of the marble is b. 11.3 m/s.
Explanation:
The velocity of a particle can be determined using the de Broglie wavelength formula: \[ \lambda = \frac{h}{mv}, \]
where \( \lambda \) is the wavelength, \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J}\cdot\text{s}\)), \( m \) is the mass of the particle, and \( v \) is its velocity.
Rearranging the formula to solve for velocity, \[ v = \frac{h}{m\lambda}. \]
Plugging in the given values (\(m = 8.66 \, \text{g} = 8.66 \times 10^{-3} \, \text{kg}\), \( \lambda = 3.46 \times 10^{-33} \, \text{m}\)), we get \[ v = \frac{(6.626 \times 10^{-34} \, \text{J}\cdot\text{s})}{(8.66 \times 10^{-3} \, \text{kg}) \times (3.46 \times 10^{-33} \, \text{m})}. \]
Calculating this expression yields approximately \( 11.3 \, \text{m/s} \).
In conclusion, the velocity of the marble is approximately ( 11.3 , text{m/s} ).
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Final answer:
The velocity of the marble is approximately 11.3 m/s. Thus the correct option is b. 11.3 m/s.
Explanation:
To find the velocity of the marble, we can use the de Broglie wavelength formula, which relates the momentum of a particle to its wavelength: λ = h / (m * v), where λ is the wavelength, h is Planck's constant, m is the mass of the particle, and v is its velocity.
Rearranging this formula to solve for velocity, we get v = h / (m * λ). Substituting the given values, we have v = (6.626 × 10^-34 m^2 kg / s) / (8.66 × 10^-3 kg * 3.46 × 10^-33 m). After calculating, the velocity of the marble is approximately 11.3 m/s.
This velocity corresponds to the particle's wave properties, which are described by wave-particle duality in quantum mechanics. The de Broglie wavelength concept suggests that particles, such as the marble in this case, exhibit wave-like behavior with a wavelength inversely proportional to their momentum.
Therefore, by determining the wavelength, we can infer the velocity of the particle. In practical terms, this principle finds applications in various fields, including particle physics and materials science, where understanding the behavior of particles at the quantum level is crucial.
By knowing the velocity, scientists can make predictions and design experiments accordingly, contributing to advancements in technology and our understanding of the universe at the microscopic level. Thus the correct option is b. 11.3 m/s.
