Question

Explain why you could or could not measure the wavelength of a golf ball in flight.

Short answer

The wavelength of a golf ball in flight is too small to measure.

Step-by-step solution

Understand the De Broglie Wavelength Concept

The De Broglie wavelength is given by the formula \( \lambda = \frac{h}{p} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), and \( p \) is the momentum of the object, defined as product of its mass and velocity \( p = mv \). It applies to particles that exhibit wave-like characteristics.

Calculate the Momentum of a Golf Ball

Assume the mass \( m \) of a standard golf ball is about 0.045 kg and its velocity \( v \) is roughly 40 m/s (typical speed of a driven golf ball). The momentum \( p \) can be calculated as \( p = m \times v = 0.045 \times 40 = 1.8 \text{ kg m/s} \).

Calculate the De Broglie Wavelength of the Golf Ball

Use the formula for De Broglie wavelength: \( \lambda = \frac{h}{p} \). Substituting the known values: \( \lambda = \frac{6.626 \times 10^{-34} \text{ Js}}{1.8 \text{ kg m/s}} = 3.681 \times 10^{-34} \text{ m} \).

Interpret the Wavelength Result

The calculated wavelength \( 3.681 \times 10^{-34} \text{ m} \) is extremely small, even smaller than atomic scales, which are typically on the order of picometers (\(10^{-12} \text{ m}\)). This makes it impractical to measure the wavelength of a macroscopic object like a golf ball.

Conclusion based on the Feasibility of Measurement

In practice, measuring wavelengths this small is beyond current technology, as we cannot detect or interpret interactions at such minuscule scales for everyday macroscopic objects. Wave-like properties are significant only for atomic and subatomic particles, not for macroscopic objects like a golf ball.

Key concepts

Momentum

To understand why measuring the wavelength of a golf ball is impractical, we first need to grasp the concept of momentum. Momentum represents the quantity of motion an object possesses and is calculated as the product of its mass and velocity. For example, a golf ball weighing 0.045 kg moving at a speed of 40 m/s has a momentum of \[ p = mv = 0.045 imes 40 = 1.8 \text{ kg m/s}. \]This calculation shows that momentum is a measure of how difficult it is to stop an object. The higher the momentum, the more energy required to change its state of motion. In quantum mechanics, momentum plays a crucial role in determining the De Broglie wavelength of an object, linking its particle-like and wave-like properties.

Wave-Particle Duality

Wave-particle duality is one of the fundamental concepts in quantum mechanics. It suggests that every object exhibits both wave-like and particle-like properties.For microscopic particles, like electrons, we often observe their wave characteristics using the De Broglie wavelength, calculated as:\[ \lambda = \frac{h}{p}, \]where \( h \) is the Planck's constant \((6.626 \times 10^{-34} \text{ Js})\) and \( p \) is the momentum. For subatomic particles, the wavelength is large enough to be observed in experiments like electron diffraction.However, this concept becomes less practical for macroscopic objects like a golf ball. The calculated De Broglie wavelength of such an object is incredibly small, denoted as \( 3.681 \times 10^{-34} \text{ m} \), making it impossible to measure or detect its wave-like properties. This demonstrates the boundary where wave-particle duality visibly affects objects, primarily at quantum scales.

Macroscopic Object Limitations

When it comes to macroscopic objects like golf balls, their wave-like characteristics are negligible. The reason lies in the sheer size of these objects. Their De Broglie wavelengths are so tiny that they cannot be detected or have any practical impact on their behavior. The wavelength calculated for a golf ball in motion, \( 3.681 \times 10^{-34} \text{ m}, \)is far smaller than atomic sizes, which makes it irrelevant for any classical or quantum interaction. Technologically, current instruments are incapable of measuring such microscopic wavelengths for large objects because they demand extremely sensitive and precise devices. This demonstrates one of the significant limitations when trying to extend quantum mechanical principles observable in smaller particles to everyday objects.