Question

How fast must a 142 - \(\mathrm{g}\) baseball travel to have a de Broglie wavelength equal to that of an \(\mathrm{x}\) -ray photon with \(\lambda=100 . \mathrm{pm} ?\)

Short answer

The speed of the baseball should be \(4.666 \times 10^{-23} \, \text{m/s} \).

Step-by-step solution

- Understand the de Broglie wavelength formula

The de Broglie wavelength for a particle is given by the formula:\[\lambda = \frac{h}{mv}\]where \(\lambda\) is the wavelength, \(h\) is the Planck constant \( (6.626 \times 10^{-34} \, \text{Js}) \), \(m\) is the mass of the particle, and \(v\) is the velocity of the particle.

- Convert units

Given the wavelength \(\lambda = 100 \, \text{pm} \), convert it to meters:\[ \lambda = 100 \, \text{pm} = 100 \times 10^{-12} \, \text{m} = 1 \times 10^{-10} \, \text{m} \]Also, convert the mass of the baseball from grams to kilograms:\[ m = 142 \, \text{g} = 0.142 \, \text{kg} \]

- Substitute known values into the formula

Using the values:\[ \lambda = 1 \times 10^{-10} \, \text{m} \], \( h = 6.626 \times 10^{-34} \, \text{Js} \), and \( m = 0.142 \, \text{kg} \), substitute them into the de Broglie formula:\[1 \times 10^{-10} = \frac{6.626 \times 10^{-34}}{0.142 \, v} \]

- Solve for velocity (v)

Rearrange the equation to solve for \( v \):\[ v = \frac{6.626 \times 10^{-34}}{0.142 \times 1 \times 10^{-10}} \]Calculate the value:\[ v = \frac{6.626 \times 10^{-34}}{1.42 \times 10^{-11}} = 4.666 \times 10^{-23} \, \text{m/s} \]

Key concepts

de Broglie wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics. It describes the wavelength associated with a particle and is named after the French physicist Louis de Broglie. According to de Broglie, every particle with momentum exhibits wave-like properties, and its wavelength can be calculated using the formula: \(\lambda = \frac{h}{mv}\), where \(\lambda\) is the wavelength, \(h\) is the Planck constant, \(m\) is the mass of the particle, and \(v\) is its velocity. This concept bridges the gap between wave and particle theories, providing a deeper understanding of the dual nature of matter.
Understanding de Broglie wavelength is crucial in fields like quantum mechanics, particle physics, and even chemistry, as it reveals the wave-particle duality of microscopic entities.

Planck constant

The Planck constant (denoted as \(h\)) is a fundamental physical constant that plays a key role in quantum mechanics. It has a value of approximately \(6.626 \times 10^{-34} \, \text{Js}\). This constant is used to describe the quantization of energy levels in atoms, the energy of photons, and is essential in the de Broglie wavelength formula.
In the context of de Broglie wavelength, the Planck constant helps to determine the wavelength associated with a particle's momentum. Remembering this constant and its significance will deepen your understanding of various quantum phenomena. As it's a very small number, its effects are only noticeable at atomic and subatomic scales.

unit conversion

Unit conversion is a critical step in solving physics problems, especially when units need to align for meaningful calculation. In the provided exercise, we convert the de Broglie wavelength from picometers (pm) to meters (m) and the mass of the baseball from grams (g) to kilograms (kg).

• 1 picometer (\text{pm}) = \(10^{-12}\) meters (\text{m})
• 1 gram (\text{g}) = 0.001 kilograms (\text{kg})

Therefore, for the given problem:
\text{Wavelength} = 100 \text{pm} = \(100 \times 10^{-12} \, \text{m} = 1 \times 10^{-10} \, \text{m}\)
\text{Mass} = 142 \text{g} = 0.142 \text{kg}
Accurate unit conversion ensures that the values in formulas correspond correctly, leading to correct results.

velocity calculation

Calculating the velocity of a particle involves rearranging the de Broglie formula to solve for velocity (\text{v}). Here's the step-by-step method:
We start with the formula: \(\lambda = \frac{h}{mv}\)
Rewriting to solve for \(v\): \(v = \frac{h}{m\lambda}\)
Substituting the known values:
\text{Planck constant}, \(h = 6.626 \times 10^{-34} \, \text{Js}\)
\text{Mass of the baseball}, \(m = 0.142 \, \text{kg}\)
\text{Wavelength}, \(\lambda = 1 \times 10^{-10} \, \text{m}\)
This gives: \(v = \frac{6.626 \times 10^{-34}}{0.142 \times 1 \times 10^{-10}}\)
Solving this, we find: \(v = 4.666 \times 10^{-23} \, \text{m/s}\)
Understanding the steps to isolate and calculate velocity is crucial, as similar steps can be applied to other formulas involving motion and energy.

particle physics

Particle physics studies the fundamental components of matter and radiation and the interactions between them. Understanding the behavior of particles at microscopic levels, such as electrons, protons, and neutrons, is essential. Key principles in particle physics include:

• Wave-particle duality: Both wave-like and particle-like properties of particles
• Quantum mechanics: The underlying framework governing particle behavior
• Symmetries and conservation laws: Rules that particles obey in reactions

The de Broglie wavelength concept is a cornerstone in quantum mechanics, revealing that particles exhibit wave properties. Investigating these principles helps us understand the behavior of matter at its most fundamental level.
It's used in technologies like electron microscopes and applications like semiconductor physics, advancing our grasp of the quantum world.