Question

What is the de Broglie wavelength (in \(\mathrm{nm}\) ) associated with a 2.5 -g Ping-Pong ball traveling at \(15 \mathrm{mph}\) ?

Short answer

The de Broglie wavelength is \(3.939 \times 10^{-23}\) nm.

Step-by-step solution

Convert Speed to Meters per Second

First, convert the speed from miles per hour (mph) to meters per second (m/s). The conversion factor is 1 mile = 1609.34 meters, and 1 hour = 3600 seconds. Calculate:\[15 \text{ mph} \times \frac{1609.34 \text{ m}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ s}} = 6.7056 \text{ m/s}\]

Convert Mass to Kilograms

Convert the mass of the Ping-Pong ball from grams to kilograms. Since 1 gram = 0.001 kilograms:\[2.5 \text{ g} \times 0.001 \text{ kg/g} = 0.0025 \text{ kg}\]

Apply de Broglie Wavelength Formula

The de Broglie wavelength \(\lambda\) is given by the formula:\[\lambda = \frac{h}{m \cdot v}\]where:- \( h \) is Planck's constant, \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \)- \( m \) is the mass in kilograms, \( m = 0.0025 \text{ kg} \)- \( v \) is the velocity in meters per second, \( v = 6.7056 \text{ m/s} \).Substitute these values into the formula:\[\lambda = \frac{6.626 \times 10^{-34}}{0.0025 \times 6.7056} \approx 3.939 \times 10^{-32} \text{ m}\]

Convert Wavelength to Nanometers

Convert the wavelength from meters to nanometers. Since 1 meter = \(10^9\) nanometers:\[3.939 \times 10^{-32} \text{ m} \times 10^9 \text{ nm/m} = 3.939 \times 10^{-23} \text{ nm}\]

State the Final Answer

The de Broglie wavelength of the Ping-Pong ball is extremely small:\(3.939 \times 10^{-23} \mathrm{nm}.\)

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes physical properties at the smallest scales, such as atoms and subatomic particles. Unlike classical mechanics, which focuses on larger, macroscopic objects, quantum mechanics delves into a micro world filled with peculiar and counterintuitive phenomena. In this realm, particles have dual properties, possessing characteristics of both particles and waves. This is central to understanding the de Broglie wavelength, which essentially assigns a wave-like property to a particle with mass and velocity.

• It introduces concepts like uncertainty, where it's impossible to know both the position and momentum of a particle with certainty.
• The de Broglie hypothesis, stemming from quantum mechanics, suggests that all matter exhibits wave-like behavior.
• Quantum mechanics also gives rise to phenomena such as superposition, where particles can exist in multiple states simultaneously.

Understanding these quantum concepts is crucial to comprehend how particles like electrons and, hypothetically, even macroscopic objects like a Ping-Pong ball could exhibit wave properties, albeit on an unimaginably small scale.

Planck's Constant

Planck's constant is a key element in the world of quantum physics, symbolized by the letter \( h \). It's a fundamental constant that relates the energy of a photon to the frequency of its electromagnetic wave. The value of Planck's constant is approximately \(6.626 \times 10^{-34} \ \text{m}^2 \text{kg/s}\). This tiny constant reveals the quantum nature of particles and waves.

• It is pivotal in calculations involving quantum scales, such as determining the energy levels of atoms.
• Planck's constant is an integral part of the formula for the de Broglie wavelength: \( \lambda = \frac{h}{m \times v} \).
• This constant shows how quantum mechanics moves away from classical phenomena, emphasizing discrete, quantized energy levels.

In our exercise, Planck's constant helps calculate the wavelength for an object, illustrating the counterintuitive yet fascinating principle that everything has a wave nature.

Unit Conversion

Unit conversion is an essential mathematical skill in physics and other sciences, allowing the accurate translation of measurements into different units. In the de Broglie wavelength exercise, we need to convert several units to perform the calculations correctly.

• Velocity is converted from miles per hour (mph) to meters per second (m/s) using the factor where 1 mile equals 1609.34 meters and 1 hour equals 3600 seconds.
• Mass is converted from grams to kilograms, with the conversion factor of 1 gram equaling 0.001 kilograms.
• The final step involves converting the wavelength from meters to nanometers, knowing that 1 meter equals \(10^9\) nanometers.

These conversions ensure that we use compatible units in the de Broglie wavelength formula, thus achieving accuracy in our results.

Velocity Conversion

In physics, velocity conversion is critical when working with different measurement units. It is often necessary to convert velocity into a standardized unit like meters per second (m/s) for consistency in calculations. For the given exercise, we begin with the velocity of a Ping-Pong ball at 15 miles per hour (mph).

• To convert mph to m/s, multiply by the conversion factors: 1609.34 meters per mile and then divide by 3600 seconds per hour. This gets us from mph to m/s.
• Using this conversion, 15 mph translates to about 6.7056 m/s, which is more suitable for scientific calculations.
• Converting to m/s is a common practice in physics as it simplifies dealing with other metric units like meters and kilograms within calculations.

Accurate velocity conversion ensures that when we use the de Broglie formula, our results reflect true physical outcomes.