Use the de Broglie relationship to determine the wavelengths of the following objects: (a) an 85-kg person skiing at 50 km/hr (b) a 10.0-g bullet fired at 250 m/s

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Convert the speed from km/hr to m/s. Use the conversion factor: 1 km/hr = 0.27778 m/s.

Use the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \), \( m \) is the mass in kg, and \( v \) is the velocity in m/s.

Substitute the mass of the person (85 kg) and the converted velocity into the de Broglie equation.

Calculate the de Broglie wavelength using the substituted values.

Interpret the result to understand the significance of the wavelength in the context of macroscopic objects.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

de Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that relates the wavelength of a particle to its momentum. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 Js), and p is the momentum of the object. This relationship implies that all matter exhibits wave-like properties, with the wavelength inversely proportional to the momentum.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity, expressed as p = mv. In the context of the de Broglie relationship, momentum is crucial because it determines the wavelength of the object. For an object with a significant mass, like an 85-kg person, the momentum will be substantial, leading to a very short wavelength that is typically imperceptible in everyday life.

Units of Measurement

Understanding units of measurement is essential for correctly applying the de Broglie relationship. In this case, the mass should be in kilograms (kg) and the velocity in meters per second (m/s) to ensure consistency with the SI units used in Planck's constant. Converting the skiing speed from kilometers per hour to meters per second is necessary for accurate calculations of the wavelength.

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